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SCHOOL OF EDUCATION 
LIBRARY 



WHITE'S SERIES OF MATHEMATICS 



A 



SCHOOL ALGEBRA 



DESIGNED FOR tJSE IN 

siair SCHOOLS and academies 



BY 



EMERSON E. J^HITE, A.M., LL.D. 

AUTHOH OF "SEBIES OF MATHEMATICS," '* ELEMENTS OF PeDAGOGT,** 

" School Management," xto. 



>;vi» 



NEW YORK . : • CINCINNATI . : • CHICAGO 

AMERICAN BOOK COMPANY 




WHITE'S MATHEMATICS 



Oral Lessons in Number 

(for teachers) 



First Book of Arithmetic 
New Complete Arithmetic 
School Algebra 
Elements of Geometry 



WHITE'S PEDAGOGICS 
Elements of Pedagogy 
School Management 



WHITE'S SCHOOL RECORDS 
New School Register 
Monthly School Record 
Teachers' Class Record 



COPTBIOHT, 1896, BT AXXKIOAK BoOK GOMFAlfT. 

I P7 



PREFACE. 



•c* 



This work is designed for use in high schools and academies, and it 
covers sufficient ground to meet fully the entrance requirements of the 
best colleges and universities. It is specially adapted to the first two 
years of the usual highnschool course. 

It has been the author's aim to prepare a school algebra that is 
pedagogically sound, as well as mathematically accurate, and thoroughly 
adequate for its place and purpose. He has kept in mind the fact 
that the great majority of the pupils who begin the study of algebra 
are between thirteen and fifteen years of age, and hence are too young 
and immature to master successfully a text-book designed for older and 
more advanced students. These young pupils have as a class a fair 
knowledge of the analytic and inductive processes of arithmetic; and 
true pedagogical principles require that this prior training be made as 
helpful as possible in their introduction to algebra. 

The manner in which these ends have been attained is partially 
indicated by the following statements : — 

1. The early introduction and practical use of the equation. The equa- 
tion, properly called *Uhe instrument of algebra," is used from the 
beginning in the solution of simple problems, thus awakening at the 
first the pupil's interest in the new study. The pupil has been using 
the informal equation throughout his entire course in arithmetic, and so 
finds no difficulty in using the equation in solving problems in advance 
of its formal treatment. The equation is also freely used in proofs and 
explanations, and in establishing data for inductions and other generali- 
zations. This use of the equation avoids those long and abstruse verbal 
statements which so baffle the young learner. 

2. The use of arithmetical approaches to algebraic processes and princi- 
ples, especially in the first half of the work. Since arithmetic deals with 
particular numbers, and algebra with both particular and general num- 
bers, the processes and principles of arithmetic afford a natural and easy 
approach to those of algebra. This conforms to two fundamental princi- 
ples of procedure in elementary teaching ; to wit, ^^from the particular 
to the general,"*^ and ^^from the known to the related unknown.''^ These 
pedagogical principles apply to the teaching of the elements of all science, 
and they are specially helpful in the pupil's introduction to algebra. 

3. The intelligent use of the inductive method. The introduction of the 
inductive method is a characteristic feature of modern arithmetic, and its 
increasing use in algebra establishes a still closer relation between the two 
studies. In the present work, the inductive method is used, when practica- 

3 



4 PREFACE. 

ble, and algebraic principles and laws are thus easily and clearly reached. 
When formal rules are given, they are placed ajfter the exercises and 
problems, as in the author*s ** Series of Arithmetics,** — an arrangement 
specially commended by the "Committee of Ten." The method of 
deductive proof is introduced gradually as the pupil advances towards 
the closing chapters, thus making him somewhat familiar with the method 
of demonstration which he will subsequently use in geometry. 

4. The immediate application of facta and principles in simple exercises 
for practice; this being a marked feature of the chapter on algebraic 
notation, often so difficult for beginners. Practice exercises not only 
help fix the facts and principles taught in the memory, but they greatly 
increase the clearness with which they are comprehended by the young 
learner. A principle that may not be fully understood from its state- 
ment is often clearly grasped in its application. 

Attention is specially called to the character of the exercises and prob- 
lems. The essential result to be attained by the pupil in the study of the 
elements of algebra is facility and accuracy in algebraic processes. 
This is necessary to all satisfactory future progress, and, to this end, there 
must be abundant and appropriate practice. Great pains have been taken 
to make the exercises and problems in this work adequate in number^ 
variety^ and grading. There has been a careful exclusion of problems 
believed to be too difficult. Such problems not only discourage the 
learner, but call for an unprofitable use of time and energy. 

Progressive teachers will be pleased to find a number of subjects and 
processes not given in the school algebras in general use. There will 
also be found throughout the work new and elegant solutions, and 
other new features of special interest and value. It is believed that 
few text-books have been prepared with greater care, or v^ith more 
earnest effort to ascertain and meet the needs of the schools. The 
result is a progressive modern algebra. 

The author gratefully acknowledges his indebtedness to Professor 
John Macnie, of the University of North Dakota, for many exercises 
and problems, and for other contributions of subject-matter ; to Professor 
M. C. Stevens, in charge of the department of mathematics in Purdue 
University, Indiana, for a critical reading of the manuscript, and for 
many helpful suggestions, including new solutions and proofs, and other 
material ; to Professor E. A, Lyman, of the University of Michigan, for 
the critical reading of the manuscript, valuable suggestions, and other 
help ; and also to several experienced teachers of algebra in high schools, 
who have rendered important assistance. 

Columbus, O. 



CONTENTS. 



aiAPTSIt 










PAOB 


L 


Introduction • • • 7 




Algebraic Equations . • • • < 




. 7 


n. 


Algebraic Notation . • • • < 




. 16 




Positive and Negative Numbers . • 






. 30 




Tiaws , 






. 33 




Equations and Problems . . « « 






. 39 


m. 


Addition and Subtraction . . • « 






43 


IV. 


Multiplication and Division . • • « 
Fractional Coefficients 






54 

. 67 




Detached Coefficients . 


> . • • « 






. 68 




Synthetic Division 


> • • • 4 






71 


V. 


Simple Equations . 


• . • « 






74 


VL 


Formulas .... 


> • . . « 


^ 




85 




Special Fonns in Multiplication . . . 






85 




Division by Binomial Factors 






. 90 


vu. 


Factoring 

Special Methods of Factoring 

General Method of Factoring Trinomials 

Factoring by Synthetic Division . 






95 
95 

108 
113 


vm. 


Common Factors and Multiples . 






119 


IX. 


Fractions , 

Reduction of Fractions 

Addition and Subtraction of Fractions . 
Multiplication and Division of Fractions 






130 
132 
139 
143 


X. 


Simple Equations containing Fractions 






162 


XI. 


Simultaneous Equations .... 


> a 




. 169 




Simple Equations with Two Unknown Numbers . 




. 169 




Equations with Three or More Unknown Number 


3 


179 


XM. 


Involution and Evolution 




. 186 




Powers . 






. 186 




Roots . . . . r . . . 






. 193 


XIII. 


Radicals 

Reduction of Radicals ..... 
Addition and Subtraction of Radicals . - . 
Multiplication of Radicals 






. 209 
. 210 
. 215 
. 216 




Division of Radicals 








. 219 



6 CONTENTS. 

CHAPTSB PAOS 

Xin. Radicals (continued). 

Involution and Evolution of Radicals • • • . 222 

Equations involving Radicals 226 

Imaginary Numbers 228 

XIV. Fractional and Negative Exponents .... 234 

XV. Quadratic Equations 239 

Incomplete Quadratics 240 

Complete Quadratics 243 

Literal Quadratics 253 

Equations Quadratic in Form 257 

XVL Simultaneous Quadratic Equations .... 265 

General Methods of Solution 265 

Special Methods 269 

XVII. Inequalities •. . . 273 

XVIII. Ratio, Proportion, Variation 277 

Ratio 277 

Proportion 281 

Variation 287 

XIX. Progressions 293 

Arithmetical Progression 293 

Geometrical Progression 300 

Harmonic Progression 307 

XX. Logarithms 309 

Principles 311 

Table with Tabular Differences 319 

Applications to Numerical Processes .... 322 

XXI. Undetermined Coefficients and Applications . . 329 

Resolution of Fractions 330 

Expansion of Fractions into Series .... 332 

Binomial Formula 334 

XXIL Determinants 340 

Determinants of the Third Order 343 

Determinants of Any Order 347 

Properties 350 

XXIII. Curve Tracing 355 

Geometrical Representation of Equations . . . 357 

Geometrical Representation of the Roots of an Equation, 362 

XXIV. Permutations and Combinations 364 

Permutations 364 

Combinations 367 

Appendix 369 

Answers . . . . 375 



ALGEBRA. 



CHAPTER L 

INTRODUCTION. 

ALGEBRAIC EQUATIONS. 

1. If a denotes a certain number, 3 a will denote 3 times 
the number ; 4 a, 4 times the number ; and so on. 

1. If a denotes a number, what will denote 5 times the 
number ? Seven times the number ? 

2. If n denotes a number, what will 5 n denote ? 8 n ? 12 n ? 

3. If a? denotes the number of feet in a rod, what will de- 
note the number of feet in 3 rods ? In 6 rods ? In 12 rods ? 

4. If a; denotes the number of bushels of apples in a barrel, 
what will denote the number of bushels in 5 barrels? In 
15 barrels ? In 20 barrels ? 

6. If a; denotes a man's age, what will denote 3 times his 
age ? 8 times his age ? f of his age ? 

2. In this introductory chapter, the signs +, — , x, -5-, and 
= have the same meaning and use as in arithmetic. 

The expression 7 + 5 denotes that 5 is to be added to 7, and 
7 — 5 denotes that 5 is to be subtracted from 7. In like manner, 
a -f- 6 denotes that the number represented by h is to be added 
to the number represented by a, and a — h denotes that the 

7 



g ALGEBRA. [§ 2. 

number represented by 6 is to be subtracted from the number 
represented by a. 

6. The sum of 3 a and 2 a is expressed by 3 a + 2 a, which 
is 5 a. In like manner express the sum of 5 a and 4 a. How 
many times a in the sum ? 

7. Express by the sign + the sum of 4® and 6x. How 
many times x in the sum ? 

8. Express the sum of 3 a;, 2 a;, and 5x, How many times 
X in the sum ? 

9. If a; denotes A's age, and Sx B's age, what will express 
the sum of their ages ? How many times x in the sum ? 

10. If x denotes the cost of a chair, 3'x the cost of a table, 
and 5 x the cost of a lounge, what will express the cost of the 
three articles ? How many times x in the cost ? 

11. An orchard contains 5 rows of pear trees, 4 rows of 
peach trees, and 7 rows of apple trees. If x denotes the num- 
ber of trees in each row, what will express the number of 
trees in the orchard ? How many times x in the number of 
trees ? 

12. A number is divided into two parts such that the greater 
part is 4 times the less. If x denotes the less part, what will 
denote the greater ? What will express the sum of the two 
parts ? How many times x in the number ? 

13. The difference expressed by 5 a — 3 a is 2 a. What is 
the difference expressed by 7 a — 4 a ? How many times a in 
the difference ? 

14. Whatisthedifferenceexpressedby 9a— 5a? 13a— 7a? 
20a-15a? 17a-9a? 21a-12a? 

15. What is the difference expressed by7aj— 4a;? 8a;— 3a;? 
12a;-7a;? 15a;-6a;? 21a;-7a;? 

16. If 5 a; denotes A's age, and 3 x denotes B's age, what will 
express the difference of their ages ? How many times x years 
19 A older than B ? 



§ 8.] INTRODUCTION. 9 

3. If 4aj = 20, 1 Xf or simply a?, is \ of 20, which is 5; and 
if X equals 5, 3 a; is three times 5, which is 16. The number 6, 
which X here denotes, is called the value of x, 

17. If 3aj = 15, what is the value of a?? Of 5aj? Of 9aj? 

18. If Sx-\-4:X = 35, what is the value of a?? Of 4a?? 

19. If 6 a? + 4 aj = 90, what is the value of a? ? 

20. If 7a?H-5a;-f 3aj = 45, what is the value of aj? 

21. If 12 a? - 7 a? = 30, what is the value of a; ? 

22. If 7 aj + 4 a? — 5 a; = 24, what is the value of a? ? 

23. If 8 a? + 2 a; — 7 a? = 21, what is the value of »? 

4. The expression 7 + 5 = 3x4 denotes that the sum of 
7 and 5 is equal to the product of 3 and 4. In like manner, 
a-\- b = c X d denotes that the sum of the numbers represented 
by a and b is equal to the product of the numbers represented 
by c and d, 

5. The equality of two numbers may be expressed by the 
sign =, which is read "equals" or "is equal to." Thus, 
2 a; + 3 aj = 25 is read " two x plus three x equals 25" 

6. An expression denoting the equality of two numbers 
is called an equation. Thus, 2aj+3a; = 25isan equation. 

7. An equation in which all of the numbers are expressed 
by figures is called an arithmetical equation. An equation in 
which one or more of the numbers is expressed by letters is 
called an algebraic equation. Thus, 8 x 5 — 16 = 6 x 4 is an 
arithmetical equation ; and 2 a? + 5 = 15, a + 6 = 20, and 
a — 6 = c, are algebraic equations. 

8. The solution of problems by means of an algebraic 
equation is called the algebraic method. 

The first step in the solution of a problem by the algebraic 
method is to state the conditions of the problem in the form 



10 ALGEBRA. [§ 9. 

of an equation^ and the second step is to find the value of the 
unknown number. 

The finding of the value of the unknown number in an 
equation is called the solution of the equation. 

9. The advantage of the algebraic method of solving prob- 
lems is best shown by its actual use in the solution of problems 
which can also be readily solved by the methods of arithmetic. 

Take, for example, this problem : 

A's age is twice B's age, and the sum of their ages is 60 
years. What is the age of each? 

Arithmetical Solution. 

By the conditions of the problem, A's age is twice B's age, 
and B's age plus twice B's age, or 3 times B's age, is 60 years. 
Hence B's age is one third of 60 years, which is 20 years; 
and A's age is twice 20 years, which is 40 years. 

Algebbaio Solution. 

Let X denote B's age ; then 2 x will denote A's age, and, by 
the conditions of the problem, we have 

aj-f 2a; = 60; 
whence 3 a; = 60 ; 

and X = 20, B's age ; 

and 2 a? = 40, A's age. 

Hence B's age is 20 years, and A's age is 40 years. 

10. In the algebraic statement of a problem, each number is 
considered as abstract. Thus, since 20 years = 20 x 1 year, x 
in the above solution represents 20, the concrete unit (1 year) 
being omitted. To express the number of years, the abstract 
value of X, when found, is considered as multiplied by the 
omitted concrete unit (1 year). 



§ 10.] INTRODUCTION. 11 

Problems. 

1. A and B together have $45^ and A has twice as much 
money as B. How much money has each ? 

Abithmetical Solution. 

Twice B's money = A's money ; 

then twice B's money + B's money = 9 46. 

Hence 3 times B^s money = $ 45 ; 

whence B^s money = | of $ 45 = $ 15, 

and A's money =2 x $ 15 = $ 30. 

Algbbsaic Solution. 

Let X = B's money ; 

then 2 a; = A's money, 

and a; + 2a; = 39; = B's and A's money. 

Hence 3 a: = 45 ; 

whence a; = J of 45 = 15, 

and 2a; = 2x 15 = 30. 

Hence A has $ 30, and B $ 15. 

Solve the next eight problems first arithmetically and then 
algebraically. 

2. A father and his son earn together $ 56 a month, and the 
father earns 3 times as much as the son. How much does 
each earn ? 

3. A man paid f 24 for a coat and vest, and the coat cost 
5 times as much as the vest. What was the cost of each ? 

4. The sum of two numbers is 42, and the greater number 
is 5 times the less. Find the numbers. 

6. Divide $36 into two parts such that the greater shall 
be 3 times the less. 

6. Cut a piece of tape 30 yards long into two pieces such 
thatlihe longer piece shall contain 5 times as many yards as 
the shorter. 



12 ALGEBRA. [§ 10. 

7. A and B tc^tlier own 150 sheep, and A owns twice as 
many sheep as B. How many sheep does each own ? 

8. The sum of two numbers is 90, and 4 times the less 
number equals the greater. . What are the numbers ? 

9. If a number be increased by twice itself, the result will 
be 60. What is the number ? 

10. The difference between two numbers is 24, and the 
greater number is 4 times the less. Wliat are the numbers? 

Let X = the less number ; 

then 4 X = the greater number, 

and 4x — X = 3x = their difference. 

Hence 3x = 24; 

whence x = 8, the less number, 

and 4x = 32, the greater number. 

11. A father's age is 3 times the age of his son, and the 
difference of their ages is 30 years. What is the age of each ? 

12. Divide a number into two parts such that the greater 
part will be 4 times the less, and their difference 81. What is 
each part ? What is the number ? 

13. Three times the cost of a saddle was the cost of a 
harness, and the harness cost $ 12 more than the saddle. What 
was the cost of each ? 

14. A school enrolls 180 pupils, and there are 20 more girls 
than boys. How many pupils of each sex in the school ? 

Let X — number of boys ; 

then X + 20 = number of girls, 

and 2 X + 20 = number of pupils. 

Hence 2x + 20 = 180. 

Subtracting 20, 2 x = 160 ; 

whence x = 80, number of boys ; 

X + 20 = 100, number of girls. 

If a number -|- 6 = 35, then the number =35 — 5, which is 30. In 
like manner if x + 6 = 35, then x = 35 — 5 = 30 ; and if x — 5 = 25, then 
X = 25 + 5 = 30. It is thus seen that a number may be added to or sub- 
tracted from both members of an equation without affecting their equality. 



§ 10.] INTRODUCTION. 18 

16. A pole 120 feet long fell and broke into two pieces, one 
piece being 30 feet longer than the other. What was the 
length of each piece ? 

16. The sum of two numbers is 120, and their difference is 
20. What are the numbers ? 

17. Divide $ 1800 between two persons, giving to one $ 650 
more than to the other. 

18. A man bought a watch and chain for $ 85, and the cost 
of the watch was $ 5 more than 3 times the cost of the chain. 
What was the cost of each ? * 

19. In a certain election 364 votes were polled by the two 
parties, and one party had 48 majority. How many votes 
were cast by each party ? 

20. In a certain village, containing 327 persons, there are 
15 more women than men, and twice as many children as there 
are men and women together. How many of each in the 
village ? 

21. Three men. A, B, and C, bought a mill for $ 12,000. A 
paid twice as much as B, and paid 5 times as much as B. 
How much did each pay ? 

22. Divide a piece of cloth containing 42 yards into three 
pieces, making the second piece 3 times the length of the 
first, and the third piece one half of the length of the other 
two pieces together. 

23. Cut a cord 45 feet long into two pieces such that one 
piece shall be 15 feet longer than the other. 

24. Divide $8400 among A, B, and C, giving to B twice as 
much as to A, and to C twice as much as to B. 

25. Divide $18 among A, B, and C, giving to A twice as 
much as to B, and to C twice as much as to A and B together. 

26. Divide $ 1800 among three persons, giving to the second 
$ 200 more than to the first, and to the third $ 200 more than 
to the second. 



14 ALGEBRA. [§ la 

27. A is twice as old as B, and B is 15 years younger than 
C, and the sum of their ages is 95 years. How old is each ? 

28. Four times a certain number is 45 more than the num< 
ber. What is the number ? 

29. Divide $140 between two men, giving one $20 more 
than the other. 

30. The sum of A's and B's ages is 70 years, and A's age is 
4 times B's age. What is the age of each ? 

31. A school enrolls 240 pupils, and twice the mmiber of 
boys equals the number of girls. How many of each are 
enrolled ? 

32. A man sold a horse and a buggy for $ 180, and received 
twice as much for the horse as for the buggy. What was the 
price of each ? 

33. A mother is 3 times as old as her daughter, and the 
difference of their ages is 30 years. How old is each ? 

34. Divide $ 125 between A and B so that A shall receive 
$ 45 more than B. 

35. The sum of two numbers is 75, and their difference is 
15. What are the numbers ? 

36. A man owns two farms which together contain 200 
acres, and the larger farm contains 42 acres more than the 
smaller. How many acres in each farm ? 

37. A farmer who owned a flock of sheep bought 3 times as 
many sheep as he had, and then had 248 sheep. How many 
sheep did he buy ? 

38. A tree 90 feet long was broken by the wind, and the 
part left standing was 20 feet shorter than the part broken off. 
What was the length of each part ? 

39. A and B are partners in business, and A's capital is 
$500 less than twice B's, and their total capital is $5500. 
How much capital has each ? 



§ 10.] INTRODUCTION. 16 

40. An estate of $ 22,050 was bequeathed to a widow and 
two sons. The sons received equal shares, and the widow 
twice as much as the two sons together. How much did each 
receive ? 

41. Divide a line 64 inches long into two parts such that the 
longer shall be 8 inches less than twice the shorter. 

42. A banker paid f 102 in ten-dollar, five-dollar, and two- 
dollar bills, using the same number of bills of each kind. How 
many bills of each kind did he use ? How many bills in all ? 

43. A school, enrolling 76 pupils, is divided into three 
classes. There are twice as many pupils in the second class 
as in the first, and 16 more pupils in the third class than in 
the second. How many pupils in each class ? 

44. A father is twice as old as his son, and the sum of their 
ages less 12 years is 60 years. How old is each ? 

45. A mother is 3 times as old as her daughter less 10 
years, and the sum of their ages is 50 years. How old is 
each? 

46. A's age is twice B's, and B's age is twice C's, and the 
sum of all their ages is 126 years. What is the age of each ? 

47. A father and his two sons earn $140 a month, and the 
father earns twice as much as the elder son, and the elder son 
twice as much as the younger. How much does each earn ? 

48. At an election there were three candidates. The first 
received 40 votes more than the second, and 65 votes more 
than the third, and the whole number of votes cast was 306. 
How many votes did each candidate receive ? 



16 ALGEBRA. r§ H* 



CHAPTER IL 

ALGEBRAIC NOTATION. 

SYMBOLS REPRESENTING NUMBERS. 

11. In arithmetic, numbers are represented by words and by 
the Arabic numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, called figures ; 
and the numbers thus represented are definite known numbers. 
Thus, the word eight and the figure 8 alike represent a definite 
number of ones or units. 

The figure 0, called zero, denotes the absence of number. 

In algebra, numbers are represented by figures and by letters. 
Figures in algebra, as in arithmetic, represent definite known 
numbers, while letters represent any numbers, whether known 
or unknown. 

Figures are called arithmetical symbols of numbers; and 
letters, general or algebraic symbols. 

12. Known numbers, when expressed by letters, are usually 
represented by the^rs^ letters of the alphabet; as a, b, c, etc. 

13. Unknown numbers, that is, numbers whose values are 
not given or determined, are usually represented by the final 
letters of the alphabet ; as a;, y^ z. 

This distinction in the use of the first and the final letters of the 
alphabet is not always observed. Any letter may be used to represent 
an unknown number ; and numbers may be represented by the letters of 
any alphabet. 

In the Roman notation, numbers are represented by the letters 
/, F, X, L, C, D, My which denote definite known numbers, the same 
as figures. 



§ 18.] ALGEBRAIC NOTATION. 17 

14. The term quxirvtity is used in algebra as synonymous 
with number, A number expressed by letters, or by letters 
and figures in combination, may be called an algebraic quantity 
or an algebraic number. In this treatise the term number 
instead of quantity ^ is generally used. 

SIGNS OF OPERATION. 

15. In both arithmetic and algebra the operations to be 
performed with numbers are indicated by characters, called 
signs. 

16. The sign +, called plus, indicates that the number after 
the sign is to be added to the number before it. Thus, 7 -f- 5, 
read " 7 plus 5," denotes that the number 5 is to be added to 
the number 7 ; and a-\-b denotes that the number repre- 
sented by 6 is to be added to the number represented by a, or, 
more briefly, that the number b is to be added to the number a. 

17. The sign — , called minus, indicates that the number 
after the sign is to be subtracted from the number before it. 
Thus, 7 — 5, read " 7 minus 5," denotes that the number 5 is to 
be subtracted from the number 7 ; and a — b denotes that the 
number represented by b is to be subtracted from the number 
represented by a, or, more briefly, that the number b is to be 
subtracted from the number a. 

The addition of a and b is denoted by a + 6, and the sub- 
traction of b from a by a — 6. 

1. Express the addition of x and y\ 2x and 3y ; 2 and y, 

2. Express the addition of a and 2 6; 2 a and 36; a, 2b, 
and 15 ; 3 a, 2 6, and 5 c; 5, 4 a, and 6 6. 

3. Express the subtraction of y from a; x from 3y; 2 
from X] X from 2; 3y from 5x. 

18. The sign x , called the sign of multiplication, and read 
"multiplied by" or "times," indicates that the two numbers 

white's alo. — 2 



18 ALGEBRA. [§ 19. 

between wbicli it is placed are to be multiplied together. 
Thus, a X b denotes that the number a is to be multiplied by 
the number b, or the number b by the number a. The num- 
bers multiplied together are called factors; and the result 
obtained by multiplying together the factors, the product. 

19. When one or both of the numbers to be multiplied are 
represented by letters, their multiplication may be expressed 
(1) by the sign x between them, as a x & or 3 x 5 ; (2) by a 
dot between the numbers, as a • 6 or 3 • 6 ; or (3), as is more 
common, by writing the numbers together, os ctb ot Sb, 

When numbers are expressed by figures only, their mul- 
tiplication is indicated by the sign x or the dot, as 12 x 5 
or 12 . 5. 

20. The sign -f-, called the sign of diyision, and read " divided 
by," indicates that the number before it is to be divided by the 
number after it. Thus, a -*- 6 denotes that the number a is to 
be divided by the number b. 

The division of one number by another may also be indicated 
by writing the dividend over the divisor in the form of a frac- 
tion. Thus, the division of a by 6 may be indicated by a -*- 6, 

or -, often written "/j. 
b 

21. In arithmetic the operations indicated by signs can actu- 
ally be performed, but in algebra many operations can only be 
indicated. Hence in algebra the signs +, — , x, and -5- each 
may indicate both an operation and its result. Thus, a + 6 
denotes the addition of a and b, and also their sum ; a — 6, 
the subtraction of b from a, and also their difference ; a xb ot 
a • 6, the multiplication of a and b, and also their product ; and 

a -f- & or -, the division of a by b, and also their quotient. 

The expression ab indicates the product of a and b, the 

sign X or • being omitted ; and usually - indicates simply the 

b 



§22.] ALGEBRAIC NOTATION. 19 

quotient of a divided by b. In arithmetic, f denotes the 
quotient of 3 by 5 as well as a fraction of a unit. 

4. Express the addition of x and y; of a, y, and z. 

5. Express the addition of a, 2 6, and 3c; of r, m, and n. 

6. Express the subtraction of n from w ; of y from a?. 

7. Express the subtraction of ab from osy] of 2 from oa;. 

8. Express the multiplication ot xby y in three ways. 

9. Express the multiplication of x, y, and z in three ways. 

10. Write without x or • the product of a and a; of a, a:, ?/; 
of 5, a,b,C', of 3, a, a;, y ; of 10, x, y, z, 

11. Write without x or • the product of 3, ab, c; of 12, x, yz. 

12. Express the division of aj by i/ in two ways. 

13. Express the division of an by 3 a^ in two ways. 

14. Let b denote the base, arid r the rate per cent, and then 
express the percentage. 

Suggestion. The rate per cent is the number of hundredths taken, 
and hence the percentage is expressed by 6 x r, or br, 

15. Let p denote the percentage, and r the rate per cent, 
and express the base. 

16. Let p denote the principal, r the rate per cent, and t 
the time in years, and express the interest. 

17. Let p denote the principal, and i the interest, and ex- 
press the amount. 

22. The factors of a number are the numbers which, multi- 
plied together, will produce it. Thus, 5 and 7 are the factors 
of 35 ; a and b, the factors of ab ; and 3, x, y, the factors of 3 xy. 

A number that contains no integral factor except itself and 1 is called 
a prime number (§ 176). 

, If one factor of a product is equal to 0, the product is equal to 0, what- 
ever may be the finite values of the other factors. 



20 ALGEBRA. [§ 23. 

23. Factors expressed by figures are called mimerical factors, 
and factors expressed by letters are called literal factors. 
ThnSy 3 is a nuiuerical factor of Sab^ and a aad b Kteral 
factors. 

24. A known factor prefixed to another factor or set of 
factors is called a coefficieot. Thus, in 5^7 5 is the coefficient 
of a ; and in 3 ax, 3 is the coefficient of cue. a may be con- 
sidered the coefficient of x in ax, and 3 a in 3 ax, 

25. When a coefficient is expressed by figures, it is called a 
munerical coefficient ; and when it contains one or more letters, 
it is called a literal coefficient. Thus, the a in ax, and the 3 a 
in 3 ax, are the literal coefficients of x. 

When an algebraic number has no numerical coefficient 
expressed, the coefficient 1 is understood. Thus, o is the 
same as la, and xy the same as 1 2^. 

26. The term coefficient means co-fa^ctor, and, in this sense of 
the term, any factor or set of factors of a product are coeffi- 
cients of the other factors. Thus, in 3 ax, 3 is the coefficient 
of ax, 3 a the coefficient of x, a the coefficient of 3 a?, and x the 
coefficient of 3 a. 

The term coefficient is, however, usually applied to the known factors, 
numerical or literal, which are placed before the other factors of a 
product, showing how many times they are taken. 

27. A power of a number is the product ebtained by taking 
the number one or more times as a factor. Thus, 25 (5 x 5) 
is the second power, or square, of 5 ; 27 (3 x 3 x 3) is the third 
power, or cube, of 3 ; and axaxaxa is the fourth power of a. 

28. The power of a number may be indicated by writing 
at the right of the number, and a little above it, a number to 
denote how many times the given number is taken as a factor. 
Thus, tlie third power of 5 is denoted by 5^ ; the fourth power 
of a, by a^ ; and the nth power of a, by a". 



§33.] ALGEBRAIC NOTATION. 21 

29. The number that is taken one or more times as a factor 
is called the base of the power ; and the number at its right, 
denoting how many times it is taken as a factor, is called the 
exponent of the power. Thus, in a^, a; is the base, and 5 the 
exponent. 

80. When an exponent is a figure, as in a^, it is called a 
numerical exponent ; and when it is a letter, as in of", it is called 
a literal exponent. 

The first power of a number is the number itself ; and hence, 
when a number has* no exponent expressed, the exponent 1 is 
understood. Thus, 6 = 6*, and x = a;\ 

31 . The exponent of a number which is expressed by two or 
more orders is written at the right of the right-hand figure. 
Thus, 25^ denotes the third power of 25 ; and 3.04*, the fourth 
power of 3.04. (f )^ denotes the square of J. 

32. The exponent of a literal number denotes the power of 
the letter only to which it is attached. Thus, a^l^ denotes 
the product of a? and 6^, and is read " a square h cube ; " and 
3aV denotes the product of 3, a*, and a^, and is read "3a 
fourth X cube." 

33. The reciprocal of a number is unity divided by the 

number. The reciprocal of 5 is - ; and of a, — 

5 a 

18. Write the prime factors (omitting 1) of 10; of 42 j of 
11 aa;; of 6aVy; of 5ax^y^; of 21 a^h^xy^, 

19. Write the numerical coefficient of 5a^; of 3.i^; of 
7a:*^; of ax^i^] of 3a2f^; of f aa^^; of a^&V. 

20. Write the exponent (including 1) of each factor in aWoi?) 
in 5 a^a^; in ^^aWx^\ in ^a^h(^\ in ^xj^z^, 

21. Express by exponents the square of oa? ; of abx ; of 2 axy, 

22. Express by exponents the cube of 13 ; of aft ; of 2 abx ; of J. 



22 ALGEBRA. [§ 34. 

23. Express the sum of the squares of a, h, and c; the sum 
of the squares of «, y, and z, 

24. Express the sum of the cubes of 4, a, m, and a?; the 
sum of the cubes of a, h, and 2 c. 

26. Express the sum of the first three powers of x\ the 
sum of the first five powers of x. 

26. Assume the first number to be the minuend, and express 
the difference of the squares of x and y ; of m and n ; of 3 and 
a ; of 2 a and 6 ; of 3 a; and 2 y, 

27. Express the product of the cubes of a, h, and c; of the 
squares of x, y, and z. 

28. Express in the form of a fraction the cube of a divided 
by the square of h ; the square of x divided by the cube of ay. 

34. The root of a number is one of the equal factors which 
multiplied together will produce the number, or is the number 
itself. Thus, 5 is the second or square root of 25 ; 4 is the 
third or cube root of 64 ; and a is the fourth root of a\ 

35. The root of a number is indicated by the character -y/, 
called the radical sign, with a number placed above it, as -y/, 
called the index of the root. Thus, V9 denotes the second 
or square root of 9, which is 3 ; ^125, the third or cube root 
of 125, which is 5 ; and -y/a, the fourth root of a. 

36. When no index is expressed, the sign -y/ indicates the 
square root. Thus Vl6 and ^^16 each denotes the square root 
of 16. The first root of a number is the number itself, and 
hence V5 = 5. 

The expression VST is interpreted or read " the square root 
of 81 ; " ^75 + 6, " the fourth root of the sum of 75 and 6 ; " 
^64 X 8, " the cube root of the product of 64 and 8 ; " and 
V^, " the cube root of ^7." 

29. Copy and read Va — 6 ; Va^ ; -^/a^ — b^ ; -^^214. 



§41.] ALGEBRAIC NOTATION. 23 

30. Copy and read Vfl? ; V^^; VaV ; V«* + 2^; -v/5*. 

31. What is the value, or indicated root, of -s/26 ? Of ^64 ? 
OfViS? Of</256? 

32. What is the in dicated root of V72 - 8 ? Of -J^45-18? 
Of ^/35 + 14 ? Of a/16 X 4? 

33. Express the square root of 80 ; of a6; of a*— 6^; of a^+i/^. 

34. Express the cube root of 18 ; of 15 x 8 ; of a+ ftj of aV. 

SIGNS OF RELATION. 

37. The sign =, called the sign of equality, denotes that the 
numbers between which it is placed are equal (§ 5). 

38. When the equality of two numbers is expressed by the 
sign =, the numbers are said to be equated, and the result, is 
called an equation. Thus, by equating x and a+h, we form 
the equation x = a-\- h, 

39. The sign > or <, called the sign of inequality, denotes 
that the numbers between which it is placed are unequal^ the 
opening of the sign or angle being towards the greater number. 
Thus, a > 6 denotes that a is greater than b, and a<b denotes 
that a is less than h. 

The symbol =^ denotes not equal to ; >, not greater than ; ^, not less 
than. They are sometimes used. 

40. The sign .*., called the sign of deduction, stands for h>ence 
or therefore, 

41. The sign ••• or ---, called the sign of continuation, stands 
for the words and so on. Thus, 1 -|- 4 -h 7 -f 10 ••• is read 
"1 + 4 + 7 + 10, and so on.'' 

Express the following statements by means of the proper 
signs : 

1. The sum of a and b is equal to c. 

2. The sum of x and y is equal to the product of m and n. 

3. Four times a is equal to the sum of 3 times c and twice 6. 



24 ALGEBRA. [§ 42. 

4. A certain number increased by 5 is equal to 3 times the 
number decreased by 15. 

6. The square of a number increased by 5 times the num- 
ber is equal to 94. 

6. The sum of a and b is greater than their difference. 

7. The sum of x and y is less thui their product. 

SIGNS OF AGGREGATION. 

42. The parenthesis ( ), the brackets [ ], the braces { |, 
the vinculum , and the bar | are called signs of aggrega- 
tion. 

43. The first four of these signs all denote that the numbers 
inclosed are to be treated as one number ; that is, are to be 
taken c ollect ively. Thus, 7 x (5 - 3), 7 x [5 - 3], 7 x {5 - 3}, 
and 7x5 — 3, all denote that 7 is to be multiplied by the differ- 
ence of 5 and 3 ; and each sign of aggregation may be removed 
if 2 be substituted for 5 — 3. 

44. The vinculum is used in connection with the radical 
sign, as V^S + 6. The line between the numerator and the 
denominator of a fraction also acts as a vinculum. Thus, 

o 

46. The bar denotes that the number at its right and the 

number at its left are to be multiplied together. Thus, 

a^ 3 xy indicates that 3xy—2y^-{-3y^istohe multiplied by a?, 
-2f 
+ 31/* 

Find the valuB of 

1. 12 +(7 -3). 5. 9x4-(15 + 3). 

2. 25 -(13 -6). 6. 23 +(8 -3x2). 

3. 23 -(13 + 4). 7. 8(7 4-5)-(20 + 4). 

4. 17 +(7 X 5^=^). 8. 32 - [(4 x 6)- 12]. 



§46.] ALGEBRAIC NOTATION. 25 



9. 9|6-3|-3|4-h2}. 11. 36 -15 -(5x2). 

10. (5+4)6-7^^. 12.. 8(12 -7) -4(5 -2). 

46. When the numbers within a parenthesis, or other sign 
of aggregation, are represented by letters, the operations in- 
dicated cannot always be performed, but the sign of aggregar 
tion can be removed in the manner indicated below : 

(1) a+(5 — c)=a-f 5 — c. 

(2) (a-(b + c)=a—b — c. 
\a—{b — c)= a.— b + c, 

(3) a x(b — c)= axb — axc = ab^ac. 

(4) a^ib-c)=^ 

It is to be specially noted that when the parenthetical 
expression is preceded by the sign -, as in (2) above, the 
signs -h and — within it must be changed (+ to — , and — 
to +) when the parenthesis is removed. 

The reason is obvious. The subtrahend in a — (6 + c) is & -f c, 
and, if b be first subtracted from a, c must be subtracted from 
the result. The subtrahend in a — (5 — c) is ft — c, and, if b be 
first subtracted from a, c must be added to the result, that 
is, a —(6 — c)= a — 6 + c. 

Remove the parentheses in each of the following expressions, 
and reduce the result to its simplest form : 

13. x-(x-y), 20. a(2 + 6) + 2(a + 6). 

14. 4:X'-(2x-^y). 21. (a + 6)-(a-6). 

15. x-\-y—(x-y). 22. 5(a + 6)-(3a- 26). 

16. a2-(a*-4). 23. {a-b)b -(ab -j-b^, 

17. ax -{ax -{-a), , ^24. 2(ab - b^ - (2 a - b)b. 

18. Sab-(2ab-b), 25. a^-2xy-\-f -(x^-^f), 

19. a^b-(a^b-b + 4:). 26. a^-y^-(x^-xy-\-y'^. 



28 ALGEBRA. [§ 53. 

63. A polynomial that consists of two terms, as a + 6, is 
called a binomial; and one that consists of three terms, as 
a' -f 2 oft -h &*, is called a trinomial. 

An algebraic expression that consists of only one term is also called 
simple ; and one that consists of two or more terms, compound, 

64. Like terms are algebraic terms that have the same literal 
factors. Thus, dbx^ 3 ahXy and — 8 abx are like terms. Like 
terms are also called similar terms. 

66. The dejg^ee of a term is the number of its literal factors. 
Thus, a and 5 x are each of the first degree ; 3 a^ and 3 ab, each 
of the second degree ; and 5 a^ar^, a^hx, and — 3 ca?, each of the 
fourth degree. 

56. A polynomial is homogeneous when all its terms are of 
the same degree. Thus, the polynomial 3aj^— 6a^2^+5a^-i-2/^ 
is homogeneous, each term being of the third degree. 

Copy and read the following algebraic expressions; then 
give the number of terms in each, and the degree of each term : 

1. 3a6-3a2 + 6c. 7. S ax" -\- 11 a%^x -^ 5 aJ'y - 9 a^, 

2. 5a^>2 4.3a«-2a6^ 8. 3 a^a? + 5(a2 + a^^ - a^a^. 

3. 4a,-2-3a^ + 22^. ^- (^ + y)(^ -V)- ^ - ^ -f- 

4. a^-\-2xy+f, 10. 3a^~if^ + ^^. 

5. 15aV-Gaa^-h5oc^. 11. 3(a -{- b) - 5(a - b) + S ab, 

6. 4.a%-^4^-a'b\ 12. 2 a - ^^^i^ + 4(a + 6). 

3 4 

13. a^ + 4a^?/ + 6ajy + 4a;.v^ + y*. 

14. x' + 2xy-\-y^-2{x'-\-y^+a?f. 

15. 7a2c3 + 3aV-5a«c-55dL^. 

c 

16. 3 a^x-^^- ^^" ^'^ - 3(a' - a^. 

Z 4 

17. ix^y-\-3x^y^'{'Sxf + y*. 



§58,] ALGEBRAIC NOTATION. 29 

NUMERICAL VALUE. 

67. The numerical value of an algebraic expression is the 
number obtained by substituting for each letter therein its 
numerical value, and then performing the operations indicated. 
Thus, if a = 5f 6 = 4, and c = 10, the numerical value of 
4a2 - 6c is 4 X 52 - 4 X 10 = 100 - 40 = 60. 

58. In finding the numerical value of algebraic expressions, 
the following facts should be kept in mind : 

I. A term preceded by no sign is positive (+)• 
II. The coefficient 1 is understood when no other is given. 
III. A letter has the exponent 1 when no other is given. 

Let a = 5, 6 = 2, c = 3, and d = 1, in each of the following 
expressions, and then find their numerical value : 

1. 7a — 6 6. 13. 5 ac + 6 — 3 (6 -he). 

2. 5 a + 5 6c. , - /;/2 a , 6c\ , 

^*- ^ ■T■ + T)■~^^• 
3. a6— 6c. \5 3y 

4. 3a6 + 5c. 15. 4(a + 6)(c-2). 

5. abc + hc-ac. 16. 10 (a + 6 - c) - a(c - 6). 

6. a^^^c-c^. 17. 6(a2 + 62-f-c)-a262. 

7. a2+62 + c2. Ig _ah__^hc^a±c 



8. 3a6'^-562c. 



6-hc 2 6 



9! 4a-6^c^ + 5c. 19- (6^ + c^ + cP)^^. 

Co 

10. 5 06^ + 6c - 3 6cd. 20. (6 + cf -- Va -f- 6^. 

11. 4 (a + 6) +3 (a -6). 21. (a + d)' - (6 + c)l 

12. (a + 6)(a-c). 22. 6* + 2 62c2 -f- c*. 

23. (a + 6)a-6(c4-c?) -(6-c)d 

24. 4a62-6V + 4(a2-h62 + 2c + c0- 

25. (a-h26 + c)(a2-26d)-f-362c. 



30 ALGEBRA. [§ 59. 

POSITIVE AND NEGATIVE NUMBERS. 

59. In both arithmetic and algebra the signs + and — are 
used to denote respectively addition and subtraction, as already- 
shown (§§ 16, 17); but in algebra, as well as in the arts and 
sciences, these signs are also used to denote that numbers 
belong to opposite series, that is, are opposite in quality. 
When the signs -f and — are thus used, they are called 
signs of opposition, or, better, signs of quality. 

60. Numbers may be used in opposition to denote condition, 
motion, direction, time, temperature, value, etc. 

Thus, if north latitude is regarded as -f , south latitude is 
— ; if east longitude is +, west longitude is — . 

If degrees of temperature above zero are +, degrees of tem- 
perature below zero are — . 

If distance in a given direction is -f , distance in the opposite 
direction is — . 

If profit or gain is +, loss is — ; if credits are +, debits 
are — . 

61. The meaning of the signs + and — , when used to denote 
quality, may be illustrated by movements in opposite directions. 

A\ 1 1 1 1 1 1 1 \ 1 1 1 1 1 ± 1 iB 

a c 

Thus, if distance from any point in the line AB to the right, 
or towards B, is considered +, distance from any point in the 
line to the lefi, or towards A, is considered — . 

If, for example, we start at a and move 8 spaces towards B, 
the distance is denoted by -f 8 ; but if we begin at a or at c 
and move 5 spaces towards A, the distance is denoted by — 5. 

Thus, if a man starts at a given point and walks in a straight 
line to the right 8 miles, and then, turning, walks back 5 



§62.] ALGEBRAIC NOTATION. 31 

miles, the distance walked from the starting point is denoted 
by + 8 miles, and the return distance by — 5 miles, and his 
distance from the starting point is expressed by + 8 + (— 5) 
= +8 — 5=+ 3. He is 3 miles to the right of the starting 
point. 

But if, on turning back, the man should walk 12 miles to 
the left, the return distance would then be denoted by — 12 
miles, and his position would be expressed by + 8 -f ( — 12) 
= 4- 8 — 12 = — 4. He would be 4 miles to the left of the 
starting point. 

1. A man starts at a given point and walks 4 miles to the 
right, and then turns and walks 7 miles to the left. Where 
is he? 

2. A ship starts at the equator and sails 50° north, and then 
turns and sails 35° south. What is its latitude ? 

3. A ship that is 30° north latitude sails 45° south. What 
is its latitude ? 

4. The temperature at noon of a certain day was 60° above 
zero, and in three hours it fell 25°. What was the temperature 
at 3 P.M. ? 

5. The temperature at 10 a.m. was 24° above zero, and at 
4 P.M. it had fallen 40°. What was the temperature at 4 p.m. ? 

6. The temperature at noon of a certain day was +30°, 
and at midnight it was — 15°. What was the difference in 
temperature ? 

62. It is thus seen that in algebra the signs + and — have 
two distinct uses : one to indicate an operation^ and the other 
to denote the quality of numbers. 

When the sign + or — is placed between two numbers or 
terms, it indicates an operation; but when the sign + or — 
is placed before a monomial or before the first term of a poly- 
nomial, it is a sign of quality. Thus, in — a6, -— d^ + he, and 
— a* + oft + 6*, the sign — denotes quality. 



32 ALGEBRA. [§6S. 

63. The sign between any two terms of a polynomial may 
"be made a sign of quality by preceding it by the sign +. 
Thus, a* — 2a6 + 6* may be written a^-f (— 2a^>) + (-|- 6*); 
that is, a^ — 2ab-{-V expresses the sum of the terms 4- a^, 
-2aby +6*. 

64. The sign + or — before a parenthesis is a sign of oper- 
ation when it is preceded by a term, as in a — (6 -|- c) ; but it 
is a sign of quality when the parenthesis stands alone, as in 

— (a — b), or is the first term of a polynomial, as in —(a —6) 
+ ab. The first term tcithin a parenthesis has the sign + or 

— as a sign of quality, as in (— a+6c). The sign + is usually 
not expressed, as in a — (6 4- c). 

65. A number preceded by the sign -f, expressed or under- 
stood, is called positive ; and a number preceded by the sign 

— is called negative. When no sign precedes an algebraic 
number, the sign + is understood (§ oS), 

66. The signs + and — are called respectively positive and 
negative. When two or more algebraic numbers have each the 
sign -f- or — , they are said to have like signs; and when one 
of two algebraic numbers is preceded by + and the other by 
— , they are said to have unlike signs. 

67. The value of an algebraic number, considered independ- 
ently of its sign, is called its absolute value. Thus, + 8 and 

— 8 have the same absolute value, but + 8 has a greater alge- 
braic value than — 8. 

The algebraic value of negative numbers decreases as their 
absolute value increases. Thus, — 10 < — 1, and — 5a < — a. 

Every algebraic number has an absolute value. The sign 
+ or — shows that this value is positive or negative. 

68. The range of algebraic numbers is double that of the 
numbers in arithmetic, since the latter have no sign of quality, 
the signs + and — being used in arithmetic to denote opera- 
tions only. 



§71.] ALGEBRAIC NOTATION. 83 

This double range of algebraic numbers is shown by the 
series : 
_8 -7 -6 -5 -4 -3 -2 -1 -0 +1 +2 +3 +4 +5 +6 +7 +8... 

The numbers on the right of in this series increase from left 
to right (4- 4 < -f 5), while those on the left of decrease from 
right to left (-4 >- 5). 

LAWS OF THE SIGNS. 

69. The manner in which positive and negative numbers 
are added is shown by the following equations: 

+ 5 + (+3) = + 5 + 3 = + 8. 
+ 2a + (+a)= + 2a + a = + 3a. 

+ 5 + (-3) = + 5-3 = + 2. 
+ 2a + (--a)= + 2a — a = + a. 

-5 + (+3) =-5 + 3 =-2. 

2 a + (+ a) = — 2 a + a = — a. 

_5 + (_3) =_5_3 =__8. 



a){ 

(2) { 

(3) { 

^ ^ (-2a + (-a)=-2a-a = -3a. 

70. These equations may be explained by taking the double 
scale of numbers, and counting as indicated by the signs. 

_8 -7 -6 -5 -4 -3 -2 -1 -0 +1 +2 +3 +4 +5 +6 +7 +8 
'  ' I I I 1 1 1 1 I I I I I I   

Thus, in (1) begin at + 5 and count 3 spaces to the right, 
to + 8, thus showing that + 5 + 3 = + 8. In (2) begin at + 5 
and count 3 spaces to the left, to +2, thus showing that 
+ 5 — 3 = + 2. In (3) begin with — 5 and count 3 spaces to 
the right, to —2, thus showing that —5 + 3= — 2. In (4) 
begin with — 5 and count 3 spaces to the left, to — 8, 'thus 
showing that -5 + (-3)=-5-3 = -8. 

71. Since the foregoing explanations are not dependent upon 
the particular numbers 5 and 3, or the relation of the numbers 

WHITENS ALO. — 3 



84 ALGEBRA. [§ 72. 

2 a and a, we may deduce the following laws of the signs in 
addition : 

I. Numbers with like signs, as in (1) and (4) in § 69, are 
added by finding the sum of their absolute values, and prefix^ 
ing the common sign to the result, 

II. Two numbers with unlike signs, as in (2) and (3), are 
added by finding the difference of their absolute values, andprefioo- 
ing the sign of the number that has the greater absolute value, 

• 

More than two numbers with unlike signs may be added by finding 
the sum of the positive numbers and the sum of the negative numbers, 
and then adding the two results, as in II. 

It is thus seen that the algebraic sum of two numbers with imlike 
signs is their arithmetical difference with the sign of the greater number. 

72. The following equations show the manner in which 
positive and negative numbers are subtracted : 

(+5-(+3) = +5-3=+2. 
X-h2a--{+a) = -{-2a — a = + a. 

+ 5-(-3)= + 5 + 3 = + 8. 

+ 2a — (— a)= + 2a + a = + 3ct. 



(1) 



(2) { 

/3x (-5-(+3)=-5-3 = -8. 

^^ (-2a-(+a)=-2a-a = -3a. 

a^ |-5-(-3)=-5 + 3=-2. 
^^ \-2a-(-a) = -2a-\-a = -a. 

73. To explain these equations, begin in (1) with -h 5 in 
§ 70, and count 3 spaces to the left, to +2, thus showing that +5 
—3= +2. In (2) begin with + 5, and since —(—3) indicates 
direction opposite to — 3, count 3 spaces to the right, to -f 8, 
thus showing that + 5 — (- 3) = + 5 + 3 = + 8. In (3) begin 
with — 6 and count 3 spaces to the left, to — 8, thus showing 
that —5—3=— 8. In (4) begin with —5, and, since — (—3) 
indicates direction opposite to — 3, count 3 spaces to the right, 
to —2, thus showing that — 5 — (— 3) = — 5 + 3 = — 2. 



§77.] ALGEBRAIC NOTATION. 35 

74. It is thus shown that numbers with like or unlike signs 
are subtracted by changing the sign of the subtraJiend, and then 
adding the resulting numbers. 

76. Since multiplication is the process of taking one num- 
ber as many times as there are units in another number, the 
multiplication of positive and negative factors may be shown 
as follows : 

( + ax(+&) = a + a-ha4---to6 terms = + a&. 

(2) (-5x(+3) = -6-6-6 = -15; 

( — ax(-f6)= — a — a — a -to 6 terms = — ab. 

Since -3 = -l-l-l and -lx8 = -8xl=-8, 

(3) f+5x(-3)=-5-5-5 = -15; 

1 + a X (—&) = — a — a — a • to 5 terms = — a6. 

Again, since -5x(-3)=-(-5)-(-5)-(-5)=+54-5+5, 

(4) (-5x(-3) = + 5 + 5 + 5 = + 15; 

( — ax(— 6)= + a + a + a+«««to& terms = + db. 

76. It is seen from (1) and (4) that the product of two 
factors with like signs is positive, and from (2) and (3) that 
the product of two factors with unlike signs is negative; that 
is, in multiplication. 

Like signs give +, and unlike signs give — . 

77. Since 5 x 3 = 15, 15 -h 3 = 6, and 15 -s- 5 = 3 ; and since 
a X b = abf ab-\-b = a, and oft -s- a = 6. It is thus seen that 
division is the inverse of multiplication, and hence the division 
of positive and negative numbers may be shown as follows ; 

(1) +ax(4-6) = a6; .-. + a6 -f- (+a) = + &. 

(2) — a X (+ 6) = — a6 ; .*. — a6 -^ (— a) = + &• 

(3) +ax(— 6)= — a6; .*. — a6 -5- (+ a) = — 6. 

(4) — ax(— 6)= + a6; .*. + oft -^ (— a) = — 6. 



86 ALGEBRA. [§ 7& 

78. it is seen from (1) and (2) in § 77 that the quotient 
obtained by the division of two numbers with like signs is posi- 
tftWf and from (3) and (4) that the division of two numbers 
with wilike Hiffns is negative; that is, in division^ 

Like slyns give -h, and unlike signs give — . 

BXBRCISBS. 

Find tlie algebraic sum of 

1. 15 and 12; 15 and - 12; - 15 and 12; - 16 and - 12. 

2. 8 + and 6 ; 8 - 6 and 5; 8 and 6 - 5; - 8 and 6 - 5; 
8 + and - 5. 

8. 12, - 7, and - 16 ; 36, - 8, - 20, and 15. 

4. 73, - 85, + 16, + 5 ; 80, - 25, - 13, - 16 ; - 20,-15, 
+ 10; 25, -13, -8, +10. 

5. 5a, ~2a, +3 a, 4a; 16a, —7a, —3a, —2a', 6a, 

— 6a, +4a, —3a, +7a. 

Make the first number given the minuend, and find the alge^ 
braic difference of 

6. 12 and +5; 12 and —5; —12 and +6; —12 and' 

— 5,5 and — 12. 

7. 23 and 8; 23 and -8; -23 and 8; -23 and -8; 

— 8 and + 23 ; - 8 and - 23. 

8. -42 and -16; 42 and +16; -42 and 16; 42 and 16; 
-16 and -42; +16 and +42. 

9. 12a and 5a; 12a and —5a; —12a and 5a', —12a 
and —5a; +5 a and — 12 a. 

10. 15a and —7a; 7a and 15a; —7a and 15a; —7a 
and — 15 a ; + 7 a and + 15 a. 



§80.] ALGEBRAIC NOTATION. 87 

Multiply 

11. 27 by -4; -27 by 4; -27 by -4. 

12. 3a by 9; 4a by -15; -3a by 12; -Sa by -13; 
-12a by 5; 12a by -3. 

13. 36a by 3; 36a by -3; -36a? by 3; - 36aj by 
-2; 36 by -2x. 

14. 6a6 by 5; 6a6 by —5; —5a6 by 12; -6a6 by -25; 
25 by -3a6; -25by4a6. 

Divide 

16. 60 by -12; -72 by 8; -84 by -12. 

16. 15aby— 5; — 15a?by— 3; — 40ajby8. 

17. 21«by3aj; 42ajby— 7aj; — 18a;by-9ic. 

18. 15aa; by —5; — 33aaj by 11; — 48aaj by — 16aa:; 
+ 48ajby -12a;. 

LAWS OF OPERATION. 

79. Algebraic operations involve several fundamental prin- 
ciples of so wide application that they are called Laws of 
Operation. 

These laws may be clearly indicated by first using particular 
numbers, represented by figures, and then using letters denoting 
general numbers. 



80. I. The Commutatiye Law. 

(7 + 3 = 

(7 x3 = 3x7 
' la X b = 



Sum. ).^--3 + 7 



Piquet. . 

b xa 



7 + 3-5 = 3-5 + 7; etc. 
a-\-b — c = b — c-{-a'y etc. 

7x3x5 = 3x5x7; etc. 
axbxc=bxcxa; etc. 

The principle thus indicated is called the Commutative Law. 
It may be stated as follows : 

The sum or the product of two or more numbers is the same 
in whomever order tJie numbers are taken. 



86 ALGEBRA. [§ 78. 

78. It is seen from (1) and (2) in § 77 that the quotient 
obtained by the division of two numbers with like signs is posi- 
tive, and from (3) and (4) that the division of two numbers 
with unlike signs is negative; that is, in division, 

Like signs give -f , and unlike signs give — . 

EXBRCISBS. 

Find the algebraic simi of 

1. 15 and 12-, 15 and - 12; - 15 and 12; - 15 and - 12. 

2. 8 4-6 and 5; 8 -6 and 5; 8 and 6 -5; -8 and 6 -5; 
8 + 6 and 6 - 5. 

3. 12, - 7, and - 16 ; 36, - 8, - 20, and 15. 

4. 73, -85, +16, +5; 80, -25, -13, -16; -20,-15, 
+ 16; 25, -13, -8, +10. 

5. 5a, —2a, +3 a, 4a; 16a, —7a, —3a, —2a\ 6a, 

— 5 a, + 4 a, — 3 a, + 7 a. 

Make the first number given the minuend, and find the alge^ 
braic difference of 

6. 12 and +5; 12 and —5; —12 and +5; —12 and' 

— 5; 5 and —12. 

7. 23 and 8; 23 and -8; -23 and 8; -23 and -8; 

— 8 and + 23; - 8 and - 23. 

8. - 42 and - 16 ; 42 and + 16 ; - 42 and 16; 42 and 16; 
-16 and -42; +16 and +42. 

9. 12a and 5a; 12a and —5a; —12a and 5a] —12a 
and —5a; + 5 a and — 12 a. 

10. 15a and —7a; 7a and 15a; —7a and 15a; —7a 
and — 15 a ; + 7 a and + 15 a. 



§80.] ALGEBRAIC NOTATION. 87 

Multiply 

11. 27 by -4; -27 by 4; -27 by -4. 

12. 3a by 9; 4a by -15; -3a by 12; -6a by -13; 
-12a by 5; 12a by -3. 

13. 36a by 3; 36a by -3; -36aj by 3; - 36aj by 
-2; 36 by -2a?. 

14. 6a6by 5; 6a& by —5; —5a6 by 12; -5a6 by— 25; 
25 by -3a6; -25by4a6. 

Divide 

15. 60 by -12; -72 by 8; -84 by -12. 

16. 15a; by -5; -15ajby-3; -40 a by 8. 

17. 21ajby3a;; 42a;by — 7a;; — 18a;by— 9a;. 

18. 15aa; by —5; — 33aa; by 11; — 48aa; by — 16aa;; 
-f- 48a; by -12a;. 

LAWS OF OPERATION. 

79. Algebraic operations involve several fundamental prin- 
ciples of so wide application that they are called Laws of 
Operation. 

These laws may be clearly indicated by first using particular 
numbers, represented by figures, and then using letters denoting 
general numbers. 



80. I. The Commutatiye Law. 

(7 + 3 = 
(a + 6 = 



Sum. ^7 + 3 = 3 + 7; 7 + 3-5 = 3-5 + 7; etc. 

& + a; a-\-b — c = b — c-\-a', etc. 

Fwduct j7x3 = 3x7; 7x3x5 = 3x5x7; etc. 



^ (7 x3 = 3x7; 7 
(ax6 = 6xa; a 



xbxc = bxcxa'y etc. 

The principle thus indicated is called the Commutative Law. 
It may be stated as follows : 

The sum or the product of two or more numbers is the same 
in whatever order tlie numbers are taken. 



8g ALGEBRA. [§ 81. 

81. It follows that the terms of a polynomial or the factors 
of a product may be arranged in any order, provided the signs 
of the terms or factors are not changed. Thus, 

^ — 2xy + f = ;ii?+f—2xy, and 3x»xy=yxa5x3. 

82. II. The AsBOCiatiye Law. 

Suyn, 
j(12 4-fi) + 7 = 12 + (r) + 7); (8 + 6)-3 = 8 + (6-3), etc. 
< (a -h ^) + c = a -f (6 + c) ; {a-\-h) — c=^a + (p — c), etc. 

Pr(Hluct 
((8x(J)x5=8x(6x5); 8x(6x6x4)=(8x6) x(5x4), etc. 
i (axh) X c=ax (b xc) ; ax{bxcxd)=(axb)xXcxd), etc. 

The principle thus indicated is called the Associatiye Law. 
It may be thus stated: 

The Hum or the product of three or more numbers is the same 
in whatever way the numbers may be grouped. 

Thifi law is the same in principle as the Commutatiye Law. 



i 



83. III. The Distributive Law. 

Product, 

[^(2 -h ;j + r>)= 3 . 2 4- 3 . 3 + 3 . 6 = 6 + 9 + 15; 

4(4 + 3 - 5) = 4 . 4 + 4 . 3 - 4 . 5 = 16 + 12 - 20 ; 

I a{h + c + rf) = a/> + ac + ac? ; {a + b — c)x = ax -{- bx — ex. 

Quotient. i±i« = i> + 12; ^ + ^ + ^ = ^ + i + g. 

3 3 3 a a a a 

The principle thus indicated is called the Distributive Law. 
It may be thus stated : 

I. TJie sum of several numbers multiplied by a given number 
equals the sum of the products of the several numbers multiplied 
by the given number. 

II. The sum of several numbers divided by a given number 
equals tJie sum of the quotients of the several numbers divided by 
the given number. 



§85.] EQUATIONS AND PROBLEMS. 89 

84. IV. The Exponent Law. 

52 =5-5; 53 = 5.5.5. .-. 52 X 5^ = 5.5. 5. 6.5 = 6* = 5*+». 

d^ z=zaa\ a? = aaa, .: a? x o? = aaaaa = a* = a*"*"*. 

a"'=aaa-.. to m factors; a" = aaa..' to n factors. Hence 
a* X a" = aaa ... to (m-\-n) factors = a'"^". 

The principle thus indicated is called the Exponent Law. It 
may be stated as follows : 

JTie exponent of the product of two or more powers of a num- 
ber is equal to the sum of the exponents of the given powers. 
Hence 

The product of the several powers of a number is found by 
adding their exponents. Thus, oi? x 0^ = x^'^^ = aJ^. 

The above law is commonly called the Index Law ; but in this treatise 
the symbol that denotes the powers of numbers is called an exponent^ and 
the symbol that denotes the roots of numbers an index. 

EQUATIONS AND PROBLEMS. 

85. The value of x found in the solution of an equation may 
be verified by substituting such value for x in the original 
equation. If the two members are equal, the value of x found 
is correct. 

Thus, in solving the equation 5a; — 2aj + 4 = 25, the value 
of X found is 7. Substituting 7 for x in the equation, we have 
5 X 7 - 2 X 7 + 4 = 25, or 35 - 14 -f-4 = 25, or 25 = 25, and 
hence 7 is the correct value of ar. 

Find and verify the value of x in the following equations : 

1. 7a; + 5a; -4a; = 24. 6. 10a;-(3a; + 2a;)= 35. 

2. 13a;-6a;4-8a;=45. 7. 12a;-(7a;- 3a;)= 32. 

3. 16a; -6a; -4a; = 30. 8. 2(9a;-f 7a;)-(3a;+5a;)=48. 

4. 20a; -12a; -3a; = 40. 9. 15a;-(4a;+6a;-3a;)=48. 
6. 8a;-f(6a;-7a;)=56. 10. 3(a; + 4) + 5 a; = 36. 



40 ALGEBRA. [§ 85. 

U. 7 a + 5(05 + 2)= 34. 16. 5(2a;--6)-4(a;-5)=37. 

12. 12a-4(a-h5)=36. 17. 6(a; - 3)+ 3(a;-f 6)= 27. 

13. 9aj-5(aj + l)=19. 18. 6(aj+4)-(3a:-f6x3)=36. 
U. 13aj-5(x*-3)=56. 19. 5(2a;-4)-(6x5~2ic)=70. 
15, 8(aj-3)-5aj-h24 = 33. 20. 4(3a:-5)+3(aj-4) = 13. 

21. A farmer raised 720 bushels of grain, consisting of 
wheat, corn, and oats. He raised twice as much wheat as 
corn, and 3 times as much oats as corn. How many bushels 
of each grain did he raise ? 

22. Divide $126 into three parts such that the second part 
shall be twice the first, and the third part 3 times the second. 

23. A market woman has 4 times as many dimes as quarters, 
and twice as many nickels as dime^, and in all she has 52 
pieces of money. How many pieces of each kind has she ? 

24. In throe years a merchant made a profit of $6300. The 
pnjfit the second year was twice the profit of the first year, 
and tlie profit the third year was twice that* of the second 
year. What was the profit each year ? 

26. Three men, A, B, and C, form a partnership in business, 
with a capital of $ 12,000. A furnished twice as much as B, 
and C as much as A and B together. How much capital did 
each furnish ? 

26. The difference of two numbers is 18, and one of the 
numbers is 3 times the other. What are the numbers ? 

27. A has $21 more than B, and A's money is 4 times B's. 
How much money has each ? 

28. A father is twice as old as his elder son, and the elder 
son is 3 times as old as the younger, and the sum of all their 
ages is 80 years. How old is each ? 

29. A newsboy, in counting his week's earnings, found that 
he had twice as many dimes as quarters, and three times as 



§ 85.] EQUATIONS AND PROBLEMS. 41 

many nickels as dimes, and that he had in all $ 7.60. How 
many pieces of money had he of each kind ? 

Let X = number of quarters ; then 26 x = their value ; 

2x = number of dimes, and 20 a: = their value ; 

6x = number of nickels, and 30 « = their value. 
Hence 25a;-f 20x + 30x= 76x = 760. 

30. A newsboy has $ 4.80 in quarters, dimes, and nickels, 
and of each an equal number. How many pieces of money 
has he? How much money in quarters, dimes, and nickels 
respectively ? 

31. A man and his two sons earn $162 a month, and the 
man earns 3 times as much as the elder son, and the elder son 
twice as much as the younger. How much does each earn ? 

32. Twice A's age is 20 years more than B's age and 10 
years more than C's age, and the sum of their ages is 120 
years. What is the age of each? 

33. A man owns two farms which together contain 180 acres 
of land, and the first farm contains 20 acres more than the 
second. How many acres in each farm ? 

34. A farm of 640 acres was divided among a brother and 
two sisters. The two sisters received an equal share, and the 
brother received 40 acres less than the number of acres received 
by the two sisters together. How many acres did each receive ? 

36. A and B together have $180, and B has $20 less than 
3 times A's money. How much money has each ? 

36. A, B, and C are partners in business. A's capital is 
twice B's, and C's capital is $ 500 less than 3 times B's, and 
the total capital is $ 14,020. How much capital has each ? 

37. Two men start at the same time from two places which 
are 63 miles apart, and travel towards each other, one at the 
rate of 3 miles an hour, and the other 4 miles an hour. In 
how many hours will they meet ? How far will each travel ? 

Suggestion. Let x = the number of hours. 



42 ALGEBRA. [§85. 

38. Three military companies muster together 195 men. 
The second company musters 30 men more than the first, and 
the third musters one half as many men as the first and second 
together. How many men in each company ? 

39. A lady bought 18 yards of silk and 15 yards of serge for 
$ 38.25, and the silk cost twice as much per yard as the serge. 
What was the cost of each per yard ? 

40. A fruit dealer sold several dozens of oranges at 25 cents 
a dozen, twice as many lemons at 15 cents a dozen, and twice 
as many pears as lemons at 10 cents a dozen, and the bill for 
all was $ 3.80. How many dozens of each kind of fruit did 
he sell ? 

41. A man bought a suit of clothes for $ 30. The coat <Jost 
$ 5 more than the trousers, and the trousers twice as much as 
the vest. What was the cost of each ? 

42. A woman bought a cloak, dress, and bonnet for $35. 
The dress cost $8 more than the bonnet, and the cloak $4 
more than the dress. What was the cost of each ? 

43. A jeweler sold three watches for $ 100. He sold the 
second watch for $ 15 more than the first, and the third watch 
for $5 less than the second. How much did he receive for 
each watch ? 

44. Two boys, Charles and Harry, had an equal amount of 
money. Charles paid 75 cents for a book, and Harry 50 cents 
for a knife, and then Harry had twice as much money as 
Charles. How much money had each at first ? 

45. A farm containing 160 acres was divided among three 
heirs, A, B, and C. A received 20 acres more than B, and C 
received as many acres as both A and B less 40 acres. How 
many acres did each heir receive? 

46. Two railroad trains start from two cities 495 miles 
apart, and run towards each other on the same track, one 
running 30 miles an hour and the other 25 miles an hour. In 
how many hours will they meet ? 



§ 88.] ADDITION. 43 



CHAPTER III. 

ADDITION AND SUBTRACTION. 
ADDITION. 

86. Addition in algebra is the process of combining two or 
more given numbers into one number, called the sum. 

Add 

1. 3 a, 5 a, 2 a, and 7 a. 

2. —2 a, —5 a, —2 a, and —7 a. 

3. ab, 4a6, 3a&, and 5ab. 

4. — a5, — 4 a6, — 6 a6, and —lab, 

5. ^a, ^a, —6a, and —la, 

6. 3 aaj, — 5 ax, — 4 a>x, and 8 oa;. 

7. a, b, and c. 

Suggestion. The sum of a, 6, and c is a + 6 + c. 

8. a, 3 h, and — d. 

9. 2 a6, 3 a6, and — 13 c. 
10. 3 aj, -a?, 4 1/, and — y. 

87. The iirst six of the foregoing examples show that like 
numbers can be combined into one term, and the last four show- 
that unlike numbers can only be added by connecting them with 
the proper sign. 

88. These examples also show that algebraic numbers may 
be 

I. Like numbers with the same sign, 

II. Like numbers vrith unlike signs, 

III. Unlike numbers, i.e., no two numbers like, 

IV. JSom^ numbers like, and others unlike. 



44 ALGEBRA. [§ 89. 

89. These facts give four classes of examples in the addi- 
tion of monomials. 

I. Like numbers with the same sign. 

Add 

II. 76c, 9&C, be, 5 be. 

12. —7 be, — 9 be, —be, — 5 be. 

13. — 3 xy, —xy, —10 xy, —4: Qcy, —7 xy, 

14. — 12 abe, — 15 abe, — abe, — 2 ahe. 

15. 3a% a% 12 a% 5a% 13 a%. 

16. -12ea?, -ea^, -2ea^, -ISex^, -9ea^. 

17. -5(a-\-b), — 6(a-h6), -3(a + &),and— (a-h&). 

18. 5xy (a — b), 3xy{a — b), 4:xy(a — b), and xy(a — b). 

19. Va&, 4 Va6, 6 Va6, 9 Va6, and 12 Vo^. 

20. -^mn^sfo^, — 3mnV?, — 9mnV^, and -wmrsfo?. 

90. To add like algebraic numbers with the same sign, 

Add the eoeffieients, and to their sum annex the literal part, 
and prefix the common sign. 

II. Like numbers with unlike signs. 

21. Add 7ab, 6ab, —2ab, - 5ab, Sab. 

For convenience write the numbers to be added in a column, 

as at the left. Add the positive numbers and the negative 

7 ab numbers, and to the arithmetical difference of the coefficients 

6a& of their sums annex the literal part, and prefix the sign of the 

-2ab greater. Thus, 7 ab -{- 6ab -{-Sab = 16ab; and - 2o6 - 6a6 

~5a6 =-7a6. 16ab -7 ab =9ab. 

^^^ It is unnecessary to write the numbers to be added in a column 

9 ^5 when the coefficients can be readily added. If preferred, the 
positive and negative monomials may be written in separate 
columns. 



§ 92.] ADDITION. 45 

Add 

22. 7 ax, —2 ax, —5 ax, 6 ox, Sax. 

23. —Sac, —ac, —5a^, — 3ac, 7 ac, 12ac. 

24. —9by, —by, 5by, by, Sby, 2by. 

25. 5 abx, — 2 abx, — 3 abx, 10 abx, — 4 dbx. 

26. Sxz^, a»*, —7x!^, 10 a»^, — as^. 

27. 8(a + c), — 3(a-f c), — ll(a + c), 7(a + c), 4(a + c). 

28. ISix-Yy), h(x^y), -\2(x^-y), 9(aj + y), -15(a:-|-y). 

29. 15 V«, Vif, — ISVi, 3VS, —5 VS. 

30. 10 a Vapy, 9 a V^, — 15 a ^xy, —7a ^sfxy, 8 a V^. 

91. To add like numbers with unlike signs, 

Add the positive numbers and the negative numbers, and to the 
arithmetical difference of their sums prefix tlie sign of the greater. 

III. Unlike numbers. 

92. In arithmetic the expression 7 + 5 indicates that the 
two numbers are to be added, and the result (12) is their sum ; 
but in algebra the expression a + 6 not only indicates that 
a and. b are to be added, but it also represents their sum (§ 21). 
Hence, when the algebraic numbers to be added are unlike, 
their sum is expressed by connecting them with their proper 
signs in the form of a polynomial. Thus, the sum of a, b, and 
— cisa + b — c. 

Every polynomial may be regarded as the expressed sum of its several 
terms. Thus, ab'^ — hc^ -f ac^ is the sum of ab^, — bc^, and ac^. 

31. What is the sum of x, y, and —z? 

32. Add Sab^, —4:a% and —5c. 

33. . Add x^, —2xy, and ^. 

34. Add 2 «*, — 3 x^y, — 4 xy^, and 5 j^. 



46 ALGEBRA. [§ 93. 

36. Add ax, bx, and — ex, and reduce the sum to its simplest 
form. 

SuooBSTioN. dx + ftas — ca5=(a + 6 — c)x. 

36. Add Saoa^ and — 9axy, and reduce to simplest form. 

93. To add unlike numbers, 

Connect the nurnbers to he added with their signs, and reduce 
the result to its simplest form. 

In algebra the results of given operations are often indicated by their 
proper signs. 

IV. Some Numbers Like, and Others Unlike. 

37. Add 15a, 36c, - 7a, - 3a, - 106c, — 2d, and 3. 

p^ In the numbers to be added, 15 a, — 7 a, 

and — 3 a are like numbers, and their alge- 
braic sum is 5 a ; Zhc and — 10 6c are like 
numbers, and their algebraic sum is — 7 &c. 
The sum sought is 5 a — 7 6c — 2 d 4- 3, 



16 a 

la 36c 
3a -106c -2d + 3 



5a— 7 6c — 2d + 3 written in the form of a polynomial. 
Add 

38. 3 a6, 5 a% 6 a6, - 8 a% - a% 6 db, and 2 ab\ 

39. — 3aa?, — 6ax^, 1 ax, — Sax, 7ax^, — a^y^, and 4aa?. 

40. 3 Vxy, — 6a^y, — 8 Vxy, 5 Vxy, and 12a^y, 

41. a^, — 2a^, — xy^, Bx^y, 4:xy^, and y^. 

42. Sanm^, — 7nx^, —5ana^, 4nar^, and lOarwc*. 

43. 15a6, — 66, — 7a6, 106, and3c. 

44. 12 a6, — 8 6c, — 5 a6, 13 6c, and — 3 a6c. 

46. 2Sx^y, -Uxy^, -llxhf, 20xf, and --3a^. 

94. To add numbers, some like and others unlike, 

Add the like numbers, and connect their suvfcs and the unlike 
numbers with their proper signs. 



§ 94.] ADDITION. 47 



Polynomials. 

1. Add 8aa — 3c2^, 3ax — 7cy, 4aaj — cy — 9a&, and 9cy 
4- 4 oft 4- aa. 

Process. 
Sax-Zcy ^j^^ ^^^ jU^g ^^^g .^ columns, as at the left, 

Sax -ley g^j^^ ^^ as in § 94. The terms of a polynomial 

4 ox — qf — vao ^^^^ ^yQ arranged in any order (Commutative Law, 

§ 80). 



ox + 9 cy 4- 4 a6 



16 oaj — 2 cy — 5 a6 

Add 

2. 5a + b-c, 2a-10 6 + 3c, 8a+76-8c, a + 26 + 6c. 

3. a + 6 + c, a — 6 + c, a + b — c, a — h — c, b — a-\'C, 

4. 4aj + 53^ — 62, 4y — 6aj4-2«, x^y — z, x + 3y + 3z. 

5. 5ar^-2^+22^ 2/^-2 ar^4-32^«*-5y^+22;2^ -3a^-f 53/*-22;2 

6. 302; — 4&y — 8, — 2a2; + 563^ + 6, 7 + 6by — 5az, 5by — 
Saz + 4,lS-Sby-^5az, 

7. 8a2^-3c2*+2, 5af-2cz^ + 5, Tcs^^-lOafS, 7a/ + 
02^ + 1, 5-2c2;2_e^2/». 

8. 19aV + 86V + 2A2, a*aj2 - 17 6y + 50%*, 12aV + 
56y-10cy, 76yT-15aW + 3A2. 

9. 5aV)c-7 aJ^c + 3 oftc^, 17 aft^c - 10 a6c«, 12 aH^c - 11 a«6c. 

10. 4a2-3c« + (?, 562-2a2 4- 0^ + ^, 2(^-762 + 5^2. 

11. 72/'-5aj*+a6c, 2iB*+9y2-3 a6c, 10 aj2+2a6c, 17 a2_ 4 y2 

12. px—qy-\-m-\-n, 5 qy— px-\-3m—Sn, 3pa5— 4gy— 4m-h7i. 

13. 21 0^-35 f +71 z", 822;2-l7aj2_43y2, 732/2_522,2_4aJJ. 

14. 18 a6 + 93 5c - 101 ac, 83 oc - 17 a5 - 23 6c, 476c- 
7a6-|-8ac, 3 a6 4- 12 oc — 32 6c. 

16. 121aj2_|-i44y2-f-492;2, 25aj*-642/»-162;*, 9y«-169«2- 
362*, 42aj2 - 26y« - 17 2^, 12 y* - 33 a* - 142*. 



48 ALGEBRA. [§ 95. 

l^x'-l^pxy, 10a^-7f. 

17. cwr" -h 6ar* -- 3 caj, 2 oar^ + 3 6aj^ — 4 caj, 7 oaj'^ + 4 ca? -- 5 6a^, 
ba^ — 5cx, 7 ex — 5 ax^, 3 ca? — 4 boc^. 

18. aj3+2ar^-f3a;+4, 6a;-5ar^+3a^-2, 3ar*-4a^+5-9a?. 

95. A polynomial with, two or more terms which have a 
common factor may be simplified by writing the factors not 
common within a parenthesis preceded or followed by the common 
factor. Thus, ax + ay — 2az may be written as a(x -^y — 2z); 
a^ — 3 ax -{- 5 bXf as (a.*^ — 3 a + 5 b)x ; and oi^ — al) + ac^, as a^ — 
a(b-(^. 

Simplify 

19. mnx^ — mnx + mny ; 3 as? — 3 oa^ — 3 axj^ — 6 ay*. 

20. Qi^ — ax — bx-\-(xc-{-ab) a^b — ah — l^d + bh. 

21. a^-3a^2^ + 5a^2/2. 5aj2_ 102^ + 6a; + 12 2r*. 

22. ax -{- by — ex — dy ', a^/x — a Vy + aV2z. 

SUBTRACTION. 

96. Subtraction is the process of taking one number from 
another. 

The result obtained by faking one number from another 
is called their difference. 

The number subtracted is called the subtrahend, and the 
number from which the subtrahend is taken is called the 
minuend. 

The difference of two numbers is found by subtracting the second from 
the first, the number first given being the minuend. Thus, the difference 
of a and 6 is a — 6. 

97. If either of two numbers be taken from their sum, the 
result will be the other number. Thus, 7 + 2 = 9; 9 — 2 = 7, 
and 9 — 7 = 2. It is thus seen that subtraction is the inverse 
of addition. 



§ 101.] SUBTRACTION. 49 

98. If the subtrahend and difference be added, their sum 
will be the minuend. It follows that the minuend may be 
regarded as the sum, and the subtrahend and difference as the 
numbers added. Hence 

Subtraction may be defined as the process of finding a num- 
ber which, added to a given number, will equal another given 
number. 

99. In arithmetic the subtrahend is equal to or less than 
the minuend; and the minuend, subtrahend, and difference 
are like numbers. 

In algebra the subtrahend may be numerically greater than 
the minuend, as 2 — 7 = — 5 ; and the minuend and subtrahend 
may be like or unlike numbers (§ 54). 

100. It has been shown in § 74 that a positive number is 
subtracted by changing its sign to — , and a negative number 
by changing its sign to -f , and then combining the resulting 
numbers as in addition. 

Monomials. 

1. From 5 a take 2 a ; from 5 a take —2 a. 

2. From —5a take 2 a ; from —5a take —2a. 

3. From 12 ab take —5ab', from — 12 ab take 7db. 

4. From —1 ax take — 3 ax ; from 7 ax take — 3 ax. 

5. From 2 xy take 10 xy ; from —2xy take — 10 ocy. 

6. From 13 aa^y tiake —5 aa^y ; from 5 aor^^ take 12 aa?]^. 

7. From x take y\ from x take — y. 

8. From 3 x take 5 ; from 2 x take — 3. 

101. To subtract one monomial from another, 

Cliange the sign of the subtrahend, and then proceed as in 
addition. 

With a little practice the sign of the subtrahend can be changed men- 
tally ; i.e., it may be considered as changed. 

white's alo. — 4 



6ab- 7aa;2+ 3' 
3a6+ 5aa;2_ 7 



50 ALGEBRA. [§ 101. 

Polynomials. 

9. From6a6 — 7aaj2-f3 take3a6-f-6aa^ — 7. 

Process, ^^^ convenience write the subtrahend under 

the minuend, as at the left. Changing the sign, 
3 ab becomes — 3 aby and —Sab and 5 ab added 
are 2 ab ; changing the sign, + 5 az^ becomes 
2 ab" 12 ax^ + 10 -6ax2, and -6ax^ and -7ax^ are -I2ax^ ; chang- 
ing the sign, — 7 becomes + 7, and + 7 and + 3 
are + 10. The difference is 2 a6 - 12 ax^ + 10. 

Proop. The sum of 2 ab - 12 ax^+ 10 and Sab+ bax^-7 ia 
6 a6 - 7 ax2 + 3. 

10. From 9 icy — 3 ar^— y take — Sxy + 5 a?— 4 y, and prove 
by addition. 

11. From a^ — 2xy + y^ take a^ -{- 2 xy + 1^, and prove by 
addition. 

12. From a-\-b take a — 6, and prove by addition. 

13. From 5a^ — Sab take a^ + a6, and prove by addition 

14. FromlOa— 56 + 3c take 8a — 76 -f 5c. 

15. From 15a^ - Tab + c take 12 a* + Sab + 4c. 

16. From9ar^ — 53^2/ — 8 take4a^ — 7a^ + 2. 

17. Yioma^ + Sa^y-\-Sxf + ftQkea^-Sichf + Sxf-y^. 

18. Fromaj«-3a^y + 3a^-2^takea^-|-3a^y + 3a^ + 3/^. 

19. From Sa^ -bab-2W take ba^-bab+l^ -10, 

20. From 7 a.-^ - 2 aaj2 + 5 a^aj - a^ take 5 o^ - 3 a^a; - a?. 

21. From a -{-b — c-\- d — e take a — 6 — c — d + e. 

22. From a — b — c — d — e take a — 6 + c — d — e. 

23. From ex? + qx—pv take a^ + pv -f- Q'^* 

24. From Ga^ + 8a6 + lOft^ take Sa^ + 6a6 + 76^ 

25. From j^ — 4p^aj — 6p;ii? + a:^ take p^ — 12p^a? + 7?, 

26. From 21 aj^ - 41 aar^ -f 72 a^ take 12 a^ - 72aaj* + 54a?. 

27. From aj* -f 4 aj^y + ^ ^2/^ + 2^ take 8 aj^ — 6 a^y* — y. 



§102.] ADDITION AND SUBTRACTION. 61 

28. From Six" -hf)-{-S(a^ - b^ take 2 (a^^-j^- 7(a« - 6^. 

29. From 6 V^T^ + Va^ - b^ + 4 V3 take Va* + 6* - V3. 

30. From aj^ — 3 a^a^y + aj* take ^ — Saba^y + y*- 

31. From aj^-Sa^y +3a^ — 2r* take a:' + 3ajy»-3/». 

102. To subtract polynomials, 

Write the subtrahend under the minuend so that like terms, if 
any, shall be in the same column. 

Change the signs of the terms of the subtrahend, and then pro- 
ceed as in addition. 

Exercises in Addition and Subtraction. 

1 . Add 3 (a: 4- y), 5(x -f y), and (x + y), 

2. Add 5-Vx — y, -\/x — y, and — 3 Va; — y. 

3. Add aix' - f), b{Q^ - y^, and 2 aio? - f). 

4. Add 3aj^ + Va — x, 5a^— Va — x, and 7 ar^ — (a — a*). 

5. From 2a; — 3Va?y 4- 2 y take a; — ^ + 2V^. 

6. From 3a^ + 2aaj + 4ar^ take a^ — oaj — ar*. 

7. From a^ + Sa^c-hxc take a;^ — (2«^c — 3 xc). 

8. Add (a — b)x, (a — b)y, and (a — b)z. 

9. Add 3a(5 — x), 2a(5 — a;)— 7a(5 — x), and a(5 — a;). 

10. From the sum of a^ -\-2xy -^ y^ and a^ — y^ take 
a* — 2a?y + 2^. 

11. From aa^ — 6a^ take the sum of 3a«* -\-5bQi:^ and — 5 oa;* 
-4.ba?. 

12. Addy*-2ar»2/^4-6, 52^ + 3a32/^ 3a^ + 2, and2^ + 2a:2r* 
+ 3a*3/* + 4aV- 

13. Add 3(a + &) + 5(a-6), 7(a + 6) - 2(a - 6), and 
12(a+6)-4(a-6). 



62 ALGEBRA. [§ 103. 

14. Add 6(ar^ - y2)^ - 5(a^ - 2/^, - 7(0^ - 2^, axid 4:(a? - f). 
16. Add 8(a2 ^b")-^ 5(0^ - b% 2(a2 +>)- 12(a2 - 6^), and 

16. Add 5(a-^b-\- c), — 7(a + 6 + c), ll(a + & + c), and 
I3(a + 6 + c). 

17. From the sum of 5c + 3aV^^, — 5 aVo(^y% and aVix^y^ 
take the sum of aVx^y^ and 3 c — 2 a^d^y^, 

18. From a^ + 3a?2t/ -f Sa^ + i/^ take a^ — ^, and from the 
result take Zo^y — 3a^^. 

Parentheses. 

103. The subtraction of one algebraic number from another 
may be indicated by writing the subtrahend in a parenthesis, 
and preceding it by the sign — . Thus, o^ -\-'if ^{x^ — 2xy -^'i^ 
indicates that ar^ — 2 a^ 4- 1/^ is to be subtracted from ar^ + 3^- 

104. If a parenthesis is preceded- by the sign +> the paren- 
thesis may be removed without changing any of the signs within 
it; but, when a parenthesis is preceded by the sign — , the signs 
4- and — within it must be changed when the parenthesis is 
removed (§ 46). Thus, ar^+i/^— (a^ — 2a^ + y^=a^ + y^ — ar^ 
+ '2xy'-f = 2xy. 

Conversely, any number of terms of a polynomial may be 
inclosed in a parenthesis preceded by the sign + without 
changing the sign of any term; but, when the terms of a poly- 
nomial are inclosed in a parenthesis preceded by the sign — , 
the signs of all the terms inclosed must be changed. Thus, 

aS _ 3a% 4- 3 aty" -h^ = aJ" -Za^b ^ (Sab^ - b^ ; 
a^ -3a^b + 3ab^ -b^ = a^-{-{-S a% - Zob^ 4- 6^. 

Remove the parenthesis and collect like terms in 

1. 5a4-&- (4a-f26). 

2. a 4-6 — c — (2a 4- 6 4- c). 

3. a2 4-2c2-(-3a2 4-2c2-ac). 



§ 104.] ADDITION AND SUBTRACTION. 63 

6. 5aj — (4a5 — 3aj + 5). 

Suggestion. First remove the vincalum, and then the parenthesis. 
It is best for the pupil to begin always with the innermost sign of 
aggregation. 



6. ic^-'(2xy+f)-(Sa^-a^-2xy-{-y^. 

7. a?— (a? — Qcy— y^^y^-\- a?*). 



8. 12-aj«-[-ic*-(aj2^aj2+5-iB^]. 

9. a' + 63 - [ (a* + 2 a6 + 6* - a^ - 6^ + 2 6*] + 4 6». 

10. aj* + a:8y + 3aj22/«-(arV + 3aj2y2 + y4). 

11. 3aaj'-2 6aj2 + 26a;-4-(-aa^-36ic* + 6a-13). 

12. aV-aa^-ar*H-(3aW + 2aa^ + iB<). 

13. a?V-(ab-h^-iroc^(a^P-ah-h^-^c). 

14. a + & + c — (a 4- & - c)-(a - 6 + c)—{c — 2a + 6). 

15. 2a-(36 + c) + [c-(a-26)], 

16. 5aj-[2^+(4ic-2^)]-(aj-2/). 

17. 3m — w— [jo— (2m — Tw)]. 

18. 8m— [4n — (4m — 4n — r)]. 

19. a-(2&-c)+3c-(46-2c)-(2a + 6 + 4c). 

20. 3aj-[-4a;+(3« + 5)-(2aj-6)]. 

21. a: + y-[(3aj-2/)+(a?-32/)-aj].4-[a?-(2/-2aj)]. 

22. 2a-[36-(4a-56)]-(6a-76). 

23. (a— 6)-{c— [d— (a + 2>)+ c] — (a + &)}. 

24. a6 — { — 6c— [a-(6 — c)— 2a6— (a-6)]|. 

26. Bx — \6y—[x—{^z — ^y)-\-2z-{6x-2y — z)']\. 

26. l-[l-a-(a2 + a-l)-(a-2a2_i)]. 

27. a-6-}-3c-[(a+2>)-(a-&-c)]-3cJ. 

28. 3[(aj - a^^) + (2 0? + 3 - 2 «)]. 



^ 



ALGEBRA. [§ l^^- 



CHAPTER IV. 

MULTIPUCATIOV AMD DIYISIOIT. 

MrXTIPUCATION. 

IW. In al-r^^ra. as in arithmetic, imiltiplicatioii is the pro- 
iv>i> of takr.iiT ^.•nt- nuuil*er as many times as there are units in 
aiivther n.:ir.l»fr. The result obtained is called the product. 

Tl.e ii\i:ul»er taken or multiplied is called the multiplicand, 
and the iinr.iWr deiK»tiiig how many times the multiplicand is 
taken is rallt^l tlie moltipUer. The multiplicand and multi- 
plier are factors of the pnxUiot 

In iirahiplio-anoTi the pnniuct is formed from the multiplicand as the 
lunhipljer is formed from unity. 

106. It has l>een shown in § 75 that a xh or —ax {—b) 
= -\-iiK and that a x {— b) or — ax6= — oft; and hence, 
when two faetors have h'ke signs, their product is positive; and, 
when two f;utors have f ml ike signs, their product is negative; 
or, nun^ briefly stated, like signs give +, and unlike signs -. 

It follows that the product of several negative factors is 
negative when the numl)er of factors is odd (1, 3, 5, 7, etc.), 
and positive when the number of factors is eveti (2, 4, 6, etc.). 

anr"law::,:;("\)><(-^)=«^x(-c)=-a^e; 

or ^a V Lm 1"""^ "" (~d)=ab xcd = abcd; 
iAf c ^ ^ (-^) X (-(f)=-a6cx (-d) = a6cd. 

X07. Since a^ — n ^ 

Vv r « ^ « a'^d «* = a X a (§ 28), 

» xa- = axaxaxaxa=a»: 
ftiul Ronorally since a" x „» _ 

of anil hiter in a product  ~ ""'^"' '* ^''^^^^ *^^* '^ exponent 
the Hcrcral factors. '* *^"*'' ^'^ *^'* ®'*'^ °f **» exponents in 

TluiB, 2 a» V 3 „s v^ „. „ 



§ 109.] MULTIPLICATION. 55 

Monomials by Monomials. 

108. 1. What is the product of 5a^b^c and Scfb<?? Of 

Ba^b^c and —3a^bc\ 

(1) ba^b^c (2) ba'^h^c Since the monomials in (1) have like 

3 a^bd^ — 3 a^bc^ signs, their product is positive ; and since 

15^6^ _ 15 a558c8 in (2) they have unlike signs, their prod- 

uct is negative (§ 106). 5 x 3 = 15 ; 
a^ X a^ = a^ ; 6^ x 6 = 6' ; c x c^ = c^. The product in (1) is 16 a^b^i^, 
and in (2) - 15 a^b^tfi. 

In multiplying two monomials, first determine the sign of 
the product, and, if —, write it; then find the product of the 
coeflS-cients, and, if not 1, write it; and lastly annex each 
letter with an exponent equal to the sum of its exponents 
in the factors. 

Multiply 

2. 3 a^xy by 5 abx. 10. — <^yV by — 11 aa^ot^t^. 

3. 12 a^bxy^ hj -3 b^ca^y^ 11. - 7 a% Vd^ by a»6 V. 

4. — 2 6a^ by —4 oan/^. 12. a^m^a^j^ hy — bVxz^. 
6. 4: a^bf hy —2 a^bcx. 13. 20 p'^g/Yt^ hy --^pqrsH. 

6. - 7 oft Vaj* by 2 ac^iB*. 14. ^dabc^xy'hy -IBa^b^xY^ 

7. 9 a^ftVy by - 10 c^bxy^, 15. - 12 ftV^^ar* by | bc^yz". 

8. — 4 oftca?* by — 7 a^ftc^on/*. 16. — 3 x^y hy —2xy hy —3 xf. 
9.-8 a^c^y^T^ by — 5 b^ch/^z. 17. — 4 a^a; by ^ oa^ by — aW. 

109. To multiply monomials. 

Multiply the numerical coefficients, and prefix the proper sign; 
then annex all the letters, giving to each an exponent equal to the 
Slim of its exponents in the several factors. 

Since a change in the order of the factors does not change 
the product (Commutative Law, § 80), for convenience arrange 
the literal factors in alphabetical order. 

If a letter occurs in only one of the factors, it will have the same 
exponent ia the product. 



56 ALGEBRA. [§ 110. 

Polynomials by Monomials. 

110. Since 3(4 + 6 -2) = 3 x 4 + 3 x 6 -3 x 2 = 24, and 

generally a(b -^ c ^ d) = ab -\- ac — ad (Distributive Law, § 83), 
it follows that a polynomial may he multiplied by a monomial 
by multiplying each term of the polynomial by the monomial, 

I. Multiply 3 a' — 5 a6 + 6 ahx by — 4a*a?. 

Process. 
3 a* _ 5 aft j_ ft ahx Write the two factors as at the left, 

__ A 2^ and multiply each term of the multipli- 

cand by — 4 a%. 

^ 12 rt*x + 20 a^bx - 24 a^hx^ 

Multiply 

2. a — 6 — c by— 5 ac; by 3 ac. 

3. x-\-y — z by —x\ by 3a;; by5ajy. 

4. a2-2a6 + 6« by 4a6; by -Soft*; by 3a%. 

5. 3ar* — 2iB2y + 5ir/ + 2/^ by — 3ic^2^; by4a?^. 

6. Sax-{-2ca? — ba^hy ^ca?^ \)j —<?x. 

7. a^b — 6a^aj — 6a^ 4- iC* by — bab^X] by 4a^. 

8. ax^ — aV — 3a^ic + c?/^ by — ax^\ by ba^oi?^. 

9. a^ftV - 76y -a;^ by -3a6a;; by 6a6V. 
10. 12/-4a;2/ + 2a;2ijy _33j22^. Ij^^3^^ 

II. a& + ^6c — c(Z by fac; by — ^acZ. 

12. ^a262 + -|a26-i-a5by|a6; by ~fa%«. 

13. xf-a^f + la^y^-fhy-^xhi^hY^x^. 

111. To multiply a polynomial by a monomial, 

Multiply successively each term of the polynomial by the mono- 
mial factor, connecting the several products with the proper signs. 



§ 113.] MULTIPLICATION. 67 



Polynomials by Polynomials. 

112. When the multiplier in arithmetic is expressed by two 
or more orders, as 45, the product is found by multiplying 
successively the multiplicand by the number denoted by each 
figure in the multiplier, and adding the partial products. Thus, 
64 X 45 = 64 X 5 + 64 X 40 = 320 + 2560 = 2880. 

In like manner in algebra, when the multiplier is a poly- 
nomial, the product is found by multiplying the multiplicand 
by each term of the multiplier, and adding the partial products, 
as shown below. 

1. Multiply a«-2a6 + 62 by a + 36. 

Process, Write the factors as at the left. Multiply 

o* — 2 a6 + 62 first each term of a^ — 2ab-h 6* by a, giving 

a -\-Sb «* — 2 a^b + ab^ as the first partial product ; next 

multiply by + 3 &, giving Sa^b-6 ab^ + 3 68 as 
the second partial product, and write the like 
terms in the same column ; lastly, add the two 



a* - 2 a^b + ab^ 

Sa^b-eab^ + Sb^ 



a' + 0^6 — 5 ab^ + 3 6* partial products. 

2. Multiply 6-3a; + 7iB2 + 8aj«byl0-8a + iB2. 

5- Sx+ 7x2+ St^ 
10- 8x+ x2 

60 -30a; +70x2 + 80 a^ 

- 40x + 24x2 _ 66x8 - 64x* 

6x2- 3^.8+ 7x* + 8x6 

60 - 70x + 99x2 4. 21x8 - 67 x* + 8x6 

113. In multiplying polynomials, it is convenient to arrange 
the terms of both factors in such order that the like terms of 
the partial products shall fall in the same column. This is 
done by selecting a letter that occurs in several terms of each 
factor, and arranging the terms in both factors according to 
the powers of such letter. The letter thus selected is usually 
the leading letter in the two factors. 



68 ALGEBRA. [§ 113. 

When the powers of the selected letter increase from left 
to right (as in 2 above), the polynomials are said to be 
arranged according to the ascending powers of the leading 
letter ; and, when the powers of such letter decrease from left 
to right (as the powers of a in 1 above), the polynomials are 
said to be arranged according to the descending powers of the 
leading letter. The polynomial a^— 2a^6a;-|-3a6V-|- 6V is 
arranged according to the descending powers of a and the 
ascending powers of b and x, 

8. Multiply 3a* -f 2a - 5a*bl^ + 4a'^6 by a» - 3a»6 -f 2a. 

2 a + 3 a^ + 4 a'& — 5 a*h^ Arrange both factors according to the as- 
2 a + a^ — 3 a^b cending powers of a, as at the left. 

4. Multiply 3- 2a* + 6a? -j-aj* by 4a? -a? 4- 5. 

x> — 2x^ + 6a; + 3 Arrange the multiplicand according to the de- 

4 x' — X + 5 Bcending powers of x, as at the left. 

Multiply 

6. a? + 3xyhj X'-2y. 11. a; — 6a by aa; — 3a*. 

6. a:*-|-2a;-3by 2a;-5. 12. a^ - f hj x^ -{- f. 

7. aj-|-4y by a; + 3^. 13. Sa^-{-SayhjSx^'-3ay. 

8. 5m*-|-2n— 1 by m*— 3n. 14. a? + 2 ay + y* by a; — y. 

9. a® -h a^* by a^ — a6l 15. m*n — mn' — 3 by mn* — m*n. 
10. a^-2ab + b^hya-b. 16. ar^- 3aj + 4 by a?- 6a:. 

17. a? — 2a^ + 2^ by aj* + 2a^ + ^. 

18. 2a? — 4a?2/-|-5aj2^ — 2/^ by a? — 3a^~4y*. 

19. 5 a?a — 3 a?a* + a^a^ — xa* by 5 a^a? — 3xa\ 

20. a^-\-b^-2ab + a^-b^hja^-db. 

21. a?-3a?y + 3a^-2/3|3y ^_^2a^ + 2/*• 
22. a^-xy-\-f + x-^y-[-lhjx-\-y — l. 

23. (a^-\-a- 20) (a - 4) by a^ - 5a + 6. 

24. (f-2!f + z^(y^z)hy{y-hz)(jy-'Z). 
26. a?-3a?-f 5by a?-a?-3. 



§ 114.] MULTIPLICATION. 69 

114. To multiply polynomials, 

Arrange the terms of each polynomial a/icording to the ascend- 
ing or descending powers of the same letter. 

Multiply all the terms of the multiplicand by each term of the 
multiplier, and add the several partial products. 

NoT£. In multiplying, observe carefnlly the laws of the signs. 

EZBRCISSS. 

Multiply 

1. oj* — a; + 1 by a? + 1. 

2. l-f-a + a^ + a5^by l-oj. 

3. a^4-2a^ + 2^2ijy ^^y^ 

4. a2-2a6-f6*by a-6. 

6. 7?-\-2xy^f\yjQ^-2xy + f. 

6. a2-2a6 + 462bya2 + 2a6 + 46l 

7. 6a52.-3ajy + 2/2by3aj2H-5aJ2^-3^. 

8. l-a + a^-a^byl+a-al 

9. 6aj3 + 4aj2 + 3a. + 2 by 6aj*-4aj*. 

10. ^ — (x?y + xy^ — 'i^hjx-^y. 

11. a« + a*62 + a26* + 6*by a^-ft*. 

12. Za^-\-2V-6abhy a^-1 ah + l. 

13. a* — a^h + a6* — 6* by a + 6 — 1. 

14. a?-\-Saa?+Za^x + a^hj 7? + 2ax-\-a\ 

15. iC* — a^ + a^ — aj + 1 by a^ + a^ — a? — 1. 

16. 5aj2-7a; + lby 7ar^ + 2ajH-4. 

17. aj» + 3a; + 9by a^-3aj + 9. 

18. lla; + ar*-24-4a5*by4aj + aJ^ + 5. 

19. 10 + 2aj + 6a^ + aJ* by 2aj-6a^ + a^. 

20. aJ*-aj2y2*+2^by 32/*4-3ar/ + 3aJ*. 

21. a^ 4- 6^ + c^ — a6 — oc — 6c by a + 6 -I- c. 

22. a^-2al) + l^ + (?hYa^ + 2ab + h^-(?. 



60 ALGEBRA. [§ 115« 

28. aJ* + y* + fljy* + «"y' + ay + «Vbyx — SfL 

24. (aj* + a')(a? -\-a)hj x — a. 

26. (6 + 2 c)(6 - c) by (6 -f c)(6 - 2c). 

Find the product of 

26. a + 6 and a — 6; 05 -j- y and x^y. 

27. 2a + 6 and 2a — 6; 2aj-h3y and 2a; — 3y. 

28. ni + n and m — n ; 2 m + n and 2 m — n. 

29. x + 2,x — 2, and a? — 3. 

30. af^ — x + 1, x + 2, and a? — 1. 

8X, OJ -f- a, a; — 2 a, a? + 4 a, and a; — a. 

32. a* + 2 a6 + 6*, a -f 6, and a — 6. 

33. aj + a, ar — a, and a^ — 2 a. 

34. What is the square ofa-|-6? Of a — 6? 

36. What is the third power ofaj-fy? Ofa; — y? 
30. Find all the powers of 2 a; + c to the third. 

37. Find the fourth power of a 4- 6; of a — 6. 

38. Find the square of a' — 2 a6 + b\ 

39. Find the square ofa;-fy + 2;; of x — y — z. 

40. What is the third power of 3a -2c? Of3a? + 2y? 

41. What is the fourth power of 2 a? — 3y ? 

115. An algebraic expression may be simplified by perform- 
ing all the multiplications and divisions indicated, removing 
the parentheses, if any, and then collecting the like terms. 

Simplify 

42. 3 a(a -f 6 H- c)- 2 a(b - c). 

43. (a* 4- 2 a6 + b^)(a - 6) - 3 a(ab + b^. 

44. (a + b)(b + c) - 6 X [6 -(a - c)]. 
46. (a + by-(a-\-b)(a-b)-(2b^-'ab). 
46. 2(a«-a6-6^-(a + 6)2. 



§ 120.] DIVISION. 61 

47. (a + l)'-(a-l)'. 

48. (a^-xy-{-f)^-2xy(-oi^-f). 

49. (x^-\-xy-f)^-(3^-xy-f)\ 

50. Substitute a 4- 2 for a; in a^ — 2 ir* -f a;. 

51 Substitute a — 1 for x, and 1 — a for y, in a* + 2 ajy + y*. 

DIVISION. 

116. Division is the inverse of multiplication (§ 77). In 
multiplication two factors are given to find their product ; in 
division a product and one of its factors are given to find the 
other factor. Hence 

117. Division is the process of finding one of two factors 
when their product and the other factor are given. 

The given product is the dividend, the given factor the 
divisor, and the factor sought the quotient. 

Since 3x4 = 12, 12^3 = 4 and 12-^4 = 3; and since 
a^ X 0?:= a*, of -i-a?z= a\ and cf -i-a^ = a^, it follows, that, if 
a prodtict be divided by one of its factors, the quotient will be 
the other fo/ctor. 

Monomials by Monomials. 

118. The process of dividing one monomial by another is 
easily learned if it be kept in mind that it is the inverse bi the 
process of multiplying two monomials. 

119. In multiplying monomials, like signs give plus, and un- 
like signs give minus (§ 106); and hence the same law of the 
signs holds true in dividing one monomial by another. 

Thus, a6 -5- 6 = + a, and — a& -^ (— 6) = + a; and ab -i-{—b) 
= — a, and — a6 -»- 6 = — a (§ 77). 

120. In multiplying monomials, the product of their coeffi- 
cients is the coefficient of the product (§ 109) ; and hence, in 
dividing one monomial by another, the coefficient of the dividend 



62 ALGEBRA. [§ 121. 

is divided by the coefficiervt of the divisor, and the result is the 
coefficierU of the qtwtierU. Thus, the numerical coefficient of 
12a'y^3aW is 12-s-3 = 4. 

121. In multiplying monomials, the exponents of each letter 
are added (§ 109) ; and hence, in dividing monomials, the expo- 
nent of each letter in the divisor is subtracted from the exponent 
of the same letter in the dividend. Thus, c^-^c^^ or^"* = a'; 
and 12 a«&» -*- 3 M* = 3 o^-'ft'-^ = 3 a*b. 

1. 21 a^ft* is a product, and 7a*&* is one of its two factors. 
What is the other factor ? 

Process. 21a»&» ^7a*l^ = (21 -5- 7)(a6-^ a2)(&8 -- 6«)= 3a»6. 

The procesB may be illustrated by resolving both monomials into their 
prime factors, and then omitting the common factors ; thus, 

21 a*&» = 7 X 3 X aaaaabbb, and 7 a^b^ = 7 x adbb. 

.%21a6&»^7a«62==Z2L32L«««««^ = 3xaaa6 = 3a»6. 

7 X aabb 

122. In the division of monomials it is often convenient to 
write the dividend over the divisor in the form of a fraction, 
as below. 

2. Divide 2Sa^i/^ by -4aY- Since 28^ ( -4) = -7, 

flB^ -4- jpS = x*, y* -r- ^ = y«, 
283^2 wid 2 -?- 1 = «, then 28a^« 

It is not necessary to di- 
Proof. — 4xV X - 7xV« = 28a^«. vide « by 1 ; z may simply 

be written in the qnotient. 

3. Divide ^Sea'^a^ (1) by 14a««y; (2) by -14 a^ajy. 

^2) ^S^ = ^«'-V = 4a3x^. 



§ 125.] DIVISION. 68 

123. It is noted that y* is omitted in the qn^otient in Ex- 
ample 3^ and this may need explanation. 

Since K = f(3 121), and ^ = 1, it follows that / = Ij and, 

since 1 may be omitted without affecting a quotient, t/^ may 
also be omitted. Any letter which would appear in a quotient 
with for an exponent may be omitted, since it is equai to 1. 

4. Divide ^a^^f^ by — 14aVy*. 

By the rule for exponents (§ 121), — = a^-« = x"«. For a full dis- 
Hussion of negative exponents, see § 434. 

5. Divide -366^2/^ by -126%; by 186c»2^. 
Divide 

6. _ 63 a^^'if by - 7 ojy. 11. 182 m^nf by 2^ m*ny. 

7. 25 m'n'a^ by 5 mn^a^, 12. 284 a%V by - 71 aVc". 

8. 24 a^bca^ by - 12 a'bo^ 13. - 1728a^3^2r» by-144a^V. 

9. _ 5ep^qr by - 7p^r, 14. - 512 a^a^fz by - 64 a^fz, 
10. - 120 aV«« by - 3 aa^. 15. 343 mn^pj^ by - 49 mn^pr. 

124. To divide one monomial by another, 

Divide the coefficient of the dividend by the coefficient of the 
divisor, prefixing the sign -|- if the signs are alike, and the sign 
— if the signs are unlike. 

Annex to this coefficient each letter of the dividend, giving it 
an exponent equal to its exponent in the dividend less its ex- 
ponent in the divisor. 

Polynomials by Monomials. 

126. Since division is the inverse of multiplication, a poly- 
nomial is divided by a monomial by dividing each term of the 
given polynomial by the monomial, observing the laws of the 



64 ALGEBRA. [§ 126. 

signs. Thus, (a+6— c)a=a'-f-a6--ac, and hence (a^-|-a6— oc) 
-«-a = — I =a-|-6 — c (Distributive Law, § 83). 

Qt Qi Qi 

1. Divide 12 aV - 8 aW/ + 4 aVj/* by 4 aV. 

3 aa;2 - 2 xy2 + ^2^8^ quotient. 

If preferred, the divisor may be written under the dividend and the 
quotient at the right, after the sign =. Thus, 

12aBa^~8a^a:8y« + 4a^a;V^3^a_2a^2 + ay 
4a2a;2 *y -r «Tr 

Divide 

2. 3aj»y-3aj*2/*4-3aJ2^ by 3ajy; by -3ajy. 

3. 4aW-8aW + 12a%3-4a262 by -4a*6. 

4. 2 00^2/* — 4 oic'y' — 6 00^2 by 2aajy. 

6. 3aa:'y — 6aV2/^ + 6a^a^ — 6aV by —3 ay. 

6. 8ajV-16aj«y*-4a^ by 4ic2. 

7. 12aV-4aV + 20aV by -2a^. 

8. 25 «*^ — 20 a^a; -f 45 icyaj^ by — 5 oeyz, 

9. 60 a^&^c^ - 48 a%V + 36 a%V by 12 a6c«. 

10. 5a^yz — 14: a^z^ — 6 a:i^yh + 20 a^y^^ hj — aj^. 

126. To divide a polynomial by a monomial. 

Divide successively each term of the polynomial by the mo- 
nomial. 

Polynomials by Polynomials. 

127. The first step in the division of one polynomial by 
another is to arrange the terms of each according to the ascend- 
ing or descending powers of a common letter, if there be such, 
as illustrated below. This order of the terms is to be kept 
throughout the entire process. 



§ 127.] 



DIVISION. 



65 



1. Divide 4 a%* + a« - 6« - 4 a*6* by a? - h\ 
Arrange the terms according to the descending powers of a. 



a2-62 



a4-3a262 + 6* 



- 3 0*62 + 4 ^254 
-" 3 a*62 _|. 3 a%^ 



a26* - 6» 
0254 _ jfi 



2. Divide 6 a^V — 4aJ2/* — 4iB^y + aj* + y* by m^-ht^ 
Arrange the terms according to the descending powers of x. 



— 20^. 



aj* - 4a;8y + 6*2^2 -ixy^ 4. y4 


x2_2xyH-y2 


x*-2a;8y+ «V 


x^-2xy-{-tf^ 


-2a^ + 4a;^2_2xy8 




xV-2xy» + y* 





3. Divide 24a^2/2_32aJjy4.i5aj4_ga^^2/4 |jy 2aj 

2x-y 



-y. 



16x* - 32a% + 24x2y2 _ gxy* + 2/* 
16ic*- Sofiy 

-24x8y + 24xV* 

-24x8yH-12xV 



8x8~12x2y4.6xy2-y» 



12x2y2-8x2/» 

12x2y2_ 6x2/8 



-2x2/8 + 2/* 
- 2 X2/8 + 2/* 

4. Divide a* + a*6* 4- ft* by a^-ab + I^. 



a* + a262 4. 64 

g* - a86 4- q^ft^ 

a % - qg62 ■!. ^58 

a262 _ a68 + 54 
a262 _ a58 _}. 54 



a^ — aft + 62 It is observed that certain 



a^ + a6 + 6^ powers of a are wanting in the 
dividend. We may insert for 
the second and third terms, or 
leave blank spaces for these 
terms, as at the left. 



white's alo. — 6 



66 ALGEBRA. [§ 127. 

Divide 

6. 6aJ*-96 by 3aj-6. 

6. jB«-3a?y + 3aV-3/* by a?-3a?y + 3aJ2/"-^. 

8. af+a^y + a^ + a^^-^ict/^ + i/^hja^ + xy-^f. 

9. 10a*-27a»6 + 34a26«-18a58-86*by5a2-6a6-2&l 

10. 36 a*6* - 60 aW + 25 a^ft* by 6 a% - 6 o^. 

11. a? + 2xy+fhjX'\-y. 

12. a^ — 2ajy + 2^ by a? — y. 

13. 9a2-fl2a6 + 462 by 3a + 26. 

14. 25a?-70xy + 4:9f hj 5x^7y. 
16. 163^-144 by 4aj-12. 

16. a'* + 6' by a + b; a* — 6* by a — 6. 

17. o' + lbyaj-l-l;! — a^byl — OS. 

18. 27iB^-82/5 by 3aj*-23^. 

19. 125m«4-343 by 5m^ + 7. 

20. m^ — n* by m -h n. 

21. a* + 6* by a 4- 6 ; a* — 6* by a — &. 

22. 9fi — t^hjx + y',sfi — i^\)ja? — y^. 

23. 125aj8 + 75a^ + 15a? + l by 5a?4-l. 

24. 216iB«-216aaJ* + 72a2aj*-8a« by 6aj*-2a. 

25. aj*-17ic2 + 16 by aj2 + 5aj + 4. 

26. a^ + a^ by aj* — aa? + a*. 

27. 27a^4-125 by 3a^ + 6. 

28. a^ + ft' by a + &; a« + 5®by a* + 6l 

29. a?-ifhj a?-f\ 243aj«-l by 3aj-l. 

30. 1728 m« - 343 by 12 m^ - 7. 

31. 15aj* - 14a;8+ 25aj2 -6aj -9 by 3a?- a?- 1. 

32. 14a^-10ar»-41a?4-25a;4-15 by 7a?-5a;-3. 

33. aj*-26aj»4-6aj* + 6»-l by aj*-6a? + l. 



§130.] FRACTIONAL COEFFICIENTS. 67 

34. l+fhyl-f+y*;l-fhjl+y' + i/*. 

35. «* — 2^ by 05* + y* ; also by a? — r^. 

36. a^ 4- a?V + y* by a* -f- 7?y^ + ^. 

37* a^ + 1 by a» + 1. 

38, 1 -aj-3a^-aj* by l + 2aj4-a^. 

39, aV — 6V by oaj* — 6Y 

40, a» + 3a26+3a6* + 6^ by a2 + 2a6 + 6'. 

41, 8a^-36ar^ + 54aj-27 by 2a?-3. 
42 l + /-2y«by 2/2 + l-2t^. 

43, 6a?* + 4a^-9a^y»-3a^ + 22/* by 2a^ + 2ajy-y». 

14. 24+38aj+43a^4-34ar»4-17iC*+6ic* by 6+5a?+4x*4-3aj». 

15. af-571a^+2118aj*+13824a;+1360 by a^+5»*+36x+136. 

128. To divide one polynomial by another, 

Arrange both dividend and divisor according to the powers of 
some common letter. 

Divide the first term of the dividend by the first term of the 
divisor, and write the result a« the first term of the quotient. 

Multiply all the terms of the divisor by the first term of the 
quotient, and subtract the product from the dividend. 

If there be a remaainder, consider it as a new dividend, divide 
its first term by the first term of the divisor, and proceed as 
before. 

FRACTIONAL COEFFICIENTS. 

129. An algebraic number is integral if it has no denomi- 
nator that contains a letter ; and hence integral numbers may 
have fractional coefficients. Thus, o? — \db -\- \V\^ dji integral 
trinomial with fractional coefficients. 

130. The fundamental processes with fractional coefficients 
are the same as those with integral coefficients. 



68 ALGEBRA. [§ 131. 

1. Add ^a&*4.|6«c4- 1 and Jaft^-^ft^c-^. 

2. Add^a»+ia6-i6^ Sa^ + :^ab-{-ib^y ^a^^ab + ^V, 
and ^ a* — a6 + f 6*. 

3. From ^«— f2^ — ficy take ix-'2xy -^^y, 

4. From la^-^aft + ^ft^ take ^a^-^oft-^dl 

5. Multiply |aj2^^^^^ by i«-i. 

6. Multiply ^a2-2a + f by ^a + i. 

7. Divide ^a^- -^a^ + ^aft^ + lfts by |a2-^a5 + |6«. 

8. Divide ^a^ -^a^ + ^x - -^ by ^aJ-i. 

9. Multiply |aj2_|a;-i by ^o;- J. 

10. Multiply |a2 + a6 + |62 ijy ^^_| J 

11. Divide oj^- iaj*-ia^-iaj2 + ^ by a^-^a?-- J. 

12. Divide iaj8 + ^aj-^ by la^ + ^aj + i. 

13. Divide |a« - ^a% + ia&' + |&^ by ^a«-|a6 + i6*. 

14. Divide iiC*- If aj^H-^ by ia^ + fa? -|. 

16. DiYide \a^-\-.^xf-{-^f hj ^af-^xy + ^y'. 

DETACHED COEFFICIENTS. 

131. When two polynomials can be arranged according to 
the ascending or descending powers of a common letter, their 
product or quotient can be readily found by operating upon 
the coeflB-cients detached for the purpose, and then supplying 
in the result the proper powers of the letters, as shown below, 

1. Multiply a« + 2a2+3a4-l by a*- 2a + l. 

Detached coefficients, 1 + 2 + 3 + 1 

1 -2 + 1 
1+2+3+1 
_2-4-6-2 

1+2 + 3 + 1 

1+0+0-3+1+1 

Supplying the powers of a, a^ + a* + a^ — 3 a* + a + 1. 
The product is a^ - 3 a^ + a + 1. 



§ 131.] DETACHED COEFFICIENTS. 69 

2. Multiply a«4-3a62-268bya«-4a*6 + 36». 

1+0+3- 2 
1-4 + 0.}. 3 

1+0+3- 2 
»4_0-12 + 8 

3 + + 9-6 

1-4 + 3-11 + 8 + 9-6 

Product, a« - 4a66 + 3a*62 _ iia«&» + Saaft* + 9a6» - 66». 

3. Multiply a^4-i«+ibya^-ia:~i. 





1 + i + i 

1 + i + J 
-i-J-i 


Product, 





4. Multiply a -^ hx -\- CO? -\- da? by m — na5 — ra?. 

a+ 6+ c+ d 
m — n — r 



am + 6m + cw + dm 
— an — bn— en — dn 

— «r — br — cr — dr 

am + (6m — an) + (cm — 6n — ar) + (c2m — en — 6r) — {dn + cr) — dr 

Product, 
am + (6m — an)a5+ (cm — 6n — ar)x2 + (dm — en — 6r)a5' — (dn + cr)a:* — {ir«*. 

6. Multiplya2 + 2a6 4-6^by a2-2a6-f6l 

6. Multiply l-2i»4-3iB2 + iB3by 2-a;4-2ar*. 

7. Multiply 2iB8 - 6aj2 + 5aj -2 by 3a^-4aj-f- 5. 

8. Multiply a^ — aj^y' + 2^ by cc* -f 7?y^ -\- y*. 

9. Multiply a^ - 3a2 + 2a - 1 by itself. 

10. Multiply a^ - 4a2 + Ha -.24 by a^ -f 4a-h 5. 

11. Multiply 4a« + 3a» + 2a + 1 by 2a8 - 3a2 - 3. 

12. Multiply ^ar» + iar^ -h I a? -fi by 2«*-iaj-i. 



70 ALGEBRA. [§ 131, 

13. Multiply aa^-\-ba? + cx + dhj ma^ ^nx^-r. 

14. Find the square ofaj* — 3fic* — 6a; + 2. 

15. Find the cube of oj* — 2 a? + 1. 

16. Dividea?*-2a:* + 8aj--3 by aj» + 2a;-l. 

flc*H-0 ~2x« + 8a;-3 |ga + 2z-l 

-2ac8- x^H-Sx 
-2x8-4x24-2x 



3x2 4-6x--3 
8x« + 6x-3 

Process by Detached Coefficients. 

1 + 0-2 + 8-3 I lH-2-1 
l-i-2-1 1-2 + 3 

-2-1 + 8 x2-2x + 3, quotient 

_2-4+2 

3 + 6-3 
3 + 6-3 

17. Divide2aJ* + aj8-8a^ + 17aj-12 by 2a^-3a + 4. 

2 + 1- 8 + 17 -12 12-3 + 4 • 



2-3+ 4 1+2-3 

x^ + 2 X — 3, quotient 



4- 


12 + 17 




4- 


6 + 


8 




— 


6 + 


9- 


-12 


— 


6 + 


9- 


-12 



18. Dividea:* + 6aj»-4«*+-24a:-27 by »^4-2aj — a 

1 + + + 6-4 + 24-27 | 1+0 + 2-3 

1 + + 2-3 1 + 0-2 + 9 

0-2 + 9-4 + 24 x8-2x + 9, quotient 

-2-0-4+ 6 

9 + + 18-27 
9 + + 18-27 

19. Divide a^+-aj»-9aj»-16aj-4 by iB* + 4a + 4, 

20. Divide 4a!»-iB* + 4aj by 2a»-3«" + 2«. 



§133.] SYNTHETIC DIVISION. 71 



SYNTHETIC DIVISION. 

132. Au examination of the process of dividing one polyno- 
mial by another shows that it may be abridged by observing 
the following facts : 

I. Only the first term of the divisor is used as a divisor in 
obtaining the successive terms of the quotient. 

II. The products found by multiplying the first term of the 
divisor by the successive terms of the quotient are the same as 
the first terms of the corresponding dividends^ aivd hence these 
products may be omitted in the process, 

III. Only the first term of each successive remainder is 
divided to obtain the corresponding term of the quotient, and 
hence the other terms of the several remainders need not be found, 

IV. The signs of the terms of the several partial products 
are changed in subtracting them, and this result may be 
attained by changing the signs of all the terms of the divisor 
before multiplying them; but, since the first term of each 
partial product is omitted (see II. above), it is only necessary 
to change the signs of the rem^aining terms of the divisor, 

133. The manner in which these facts may be applied in 
abridging the division process is shown below. For conven- 
ience the divisor is written at the left in a vertical column, and 
the quotient below the dividend and partial products. 



1. Divide a*-25a« + 40a-16 by a« + 5a 

Full Process, 



-4. 



a* 


Full Process, 

a* + -25a2 + 40a-16 
a* -f 6 a' — 4a^ 


+ 5a 


-6a8-21o2 + 40o 


-4 


4a2 + 20a-16 
4o2 + 20o-16 




a^ _ 5 a H- 4, quotient. 



72 



ALGEBRA. 



[§ 133. 



a2 


Abridged Process. 
04 + -25a2 + 40a -16 


— 6a 

+ 4 


-6a«+ 4aa 
25 a2 


-20a 
-20a + 16 




a2-6a + 4 


+ 0+0 



The quotient is a^ — 5 a + 4. 

Since the first term of the divisor is used only as a divisor, 
its sign is not changed, and it is separated from the succeeding 
terms, whose signs are changed^ by a horizontal line. 

The first term of the quotient is a^-i-a^ = a% written below. 

The first partial product, omitting a*, is (— 5a4-4)xa' 
= -5a»+-4a2. 

The second term of the quotient is —5a^-i-a* = — 5af 
written below. 

The second partial product is (— 5 a 4-4)x(— 5a)=25a* 
-20 a. 

The sum of the terms in the third column is + 25 a* + 4 a* — 
25 a* = + 4 a*, and the third term of the quotient is + 4 a* -*- a* 
= -h 4, written below. 

The third partial product is (— 5a + 4)x4 = — 20a + 16. 

The sum of the fourth and fifth columns being each 0, the 
division is exact, and the quotient is a* — 5 a + 4. The num- 
ber of O's is one less than the number of terms in the divisor. 

It is well to indicate where the division ends by a vertical 
line, as above. 

Process by Detached Coefficients. 



1 

-5 

+ 4 


1 + 0-25 

-6+ 4 
25 


+ 40-16 

-20 

-20 + 16 


Or (2) 1 
-6 
+ 4 


1+0-25 + 40-16 
-6 + 26-20 

+ 4-20+16 




-6+ 4+ 0+ 




1-6+ 4 


+ 0+0 


1-6+ 4 



Supplying the powers of a, the quotient is a^ — 6 a + 4. 

In the second process (2) the partial products are written 
diagonally downward to the right, each term being opposite 



§ 133.] 



SYNTHETIC DIVISION. 



73 



the term of the divisor multiplied. If preferred, this form 
may be used. 

In this abridged process the several partial products or divi- 
dends are formed by addition or synthesis, and hence the 
process is called synthetic division. 

It will be showD later (§ 215) that this method is of great advantage 
in factoring polynomials that cannot otherwise be readily factored. It 
may also be used with advantage in finding the integral roots of certain 
higher equations (§ 472). 

2. Divide 2aJ* + aj8-22ic2 + 34a?- 15 hja^ + Sx-d. 



1 
-3 

+ 6 



2 + 1-22 +34-16 
-6 + 10 



+ 15 



-25 
-9+15 



2-5+ 3 I + 0+ 
The quotient is 2*2 — 5 a; + 3. 

3. Divide 2ar^+3aj*-8a^-20ar*+30a?+60 by 2aj»-5a^+ia 

2 + 3-8 -20 + 30 + 60 
+ 5+0 -10 



+ 5 
+ 
-10 



+ 20 



+ 0-40 
30+ 0-60 



1 + 4+ 6|+ 0-10- 
The quotient is x^-{-ix — 6, with — 10a remainder. 

Divide 

4. aj* — ar* — 7a^4-a?+6byic2_a;_5. 

5. aJ*-faj«-9a^-16a;-4 by aj2 4.43.4.4. 

6. aJ*4-2aj2-aj + 2 by aj2_a; + l. 

7. H-2i»2-i«842a^ by l-« + a^. 

8. aJ«-4a?* + 3aj* + 2a^ + 4aj2-15 by aj3-3 

10. ixfi-Sa^f + 3a^y^-i/hyix^-Sa^y-\-3xy^ 

11. aj*-^aj2-^aj-^ by ar* + ia; + f 



+ 5. 



-"2/». 



74 ALGEBRA. [§ 134. 



CHAPTER V. 
SIMPLE EQUATIONS. 

134. An equation is the expression of the equality of two 
numbers (§ 6). 

An algebraic equation is written by means of symbols ; but it may be 
read, or expressed orally, by words. 

135. The number that precedes the sign of equality is called 
the first member of the equation, or the left side; and the num- 
ber that follows the sign is called the second member, or the 
right side, 

136. An equation that is true only for certain particular 
values of the letters therein is called an equation of condition. 
Thus, 3 a? 4- 3 = 15, in which x equals 4 only, is an equation 
of condition. 

An equation which is true for all valines of the letters therein 
is called an identical equation, or, briefly, an identity. Thus 
the equation a{a — h) = a^ — ah is an identity, since it is true 
for all values of a and h. If, for example, a^B, and & = 4, 
then 5(5-4) =25-20; thatis,5 = 5. 

In an identical equation the sign =, called the sign of 
identity, may be used instead of the sign =. The identity 
a 4- a = 2 a is read, " a plus a is identical with 2 a." 

137. An equation which, when cleared of fractions, contains 
only the first power of the unknown number, is called an 

equation of the first degree. Thus, oa? — a? = a6, and a + - = -, 
are equations of the first degree. 



§141.] SIMPLE EQUATIONS. 75 

The eqnation - = ^ ~ is not of the first degree, for, when cleared of 
a X 

fractions, it becomes a"^ = a^ _ ab^ an equation of the second degree. 

An equation of the first degree is also called a simple 
equation. 

138. The solution of an equation is the finding of the value 
of the unknown number in it; i.e., the finding of a number 
which, substituted for the unknown number, will satisfy the 
equation. The value of the unknown number is called the 
root of the equation. 

139. The process of solving an equation involves the mak- 
ing of such needed changes in the two members as do not affect 
their equctXUy. 

140. These changes are based on the following self-evident 
truths or principles, called 

AXIOMS. 

1. Ifeqvxds he added to equals, the sums will be equal. 

2. If equals he subtracted from equals, the remainders mil be 
equal. 

3. If equals be multiplied by equals, the products will be equal. 

4. If equals be divided by equals, the quotients will be equal. 

5. Like powers of equals are equal. 

6. lAke roots of equals are equal. 

7. Generally, if the same changes be made in equals, the results 
wiU be equal. 

141. These axioms, when applied to algebraic equations, 
show that 

I. The same number may be added to or subtracted from both 
members of an equation. Thus, if 2; = 12, a; + 4 = 12 -h 4, and 
(5-4 = 12-4. 



76 ALGEBRA. [§ 142. 

II. Both members of an equation m^y he multiplied or divided 
by the same number. Thus, if a; = 12, a? x 3 = 12 x 3 ; and 
if a; = 12, aJ-^3 = 12^3. 

III. Like powers and like roots of both members of an equor 
tion are equal. Thus, if « = 4, ar^ = 16 ; if iB* = 16, V^ = Vi6. 

SOLUTION OF EQUATIONS. 

142. In solving an equation, it is advantageous to transfer 
the terms that contain the unknown number to the first mem- 
ber, and the terms that contain only known numbers to the 
second member. This is called transposing the terms. 

1. What is the value of x in the equation a;-f-4 = 16? 
In x-{-b = a? 

(1) Since a? + 4 = 16, a; + 4-4 = 16-4 (Ax. 2). .-. a = 16 
-4 = 12. 

(2) Since x + b = a, x-\-b — b = a — b. .\ x = a — b. 

It is seen that + 4 in (1), and -f 6 in (2), may be transposed 
to the second member if the sign -f be changed to — . 

2. What is the value of x in the equation a; — 4 = 12 ? 
In a; — b= a? 

(1) Since a;-4 = 12,a?-44-4 = 12 + 4 (Ax. 1). .-. a; = 12 
+ 4 = 16. 

(2) Since x — b = a,x — b-\-b=^a-\-b. .*. x = a-\-b. 

It is seen that — 4 in (1), and — 6 in (2), may be transposed 
to the second member if the sign — be changed to +. 

143. In like manner it may be shown that any term may 
be transposed from one member of an equation to the other, 
provided its sign be changed. Thus, if 5 a; — 8 = 16 -|- 3 a;, then 
5aj-3a; = 16 + 8. .-. 2 a: = 24, and a; = 12. 

144. The signs of all the terms of an equation may be 
changed, since in effect this is the same as multiplying both 
members by — 1 (Ax. 3). 



§146.] SIMPLE EQUATIONS. 77 

145. When the terms of an equation are transposed and 
the unknown terms combined in one term, the value of the 
unknown number is found by dividing both members by its 
coefficient. Thus, if 

(1) 3aj = 30, 3iB-^3 = 30^3 (Ax. 4). .•.iB = 10. 

/o\ J dx b b 

(2) ax = b, — = — .•.« = — 

a a a 

(3) (a — b)x = a + b, ^ f- = — '-—» .-. « = — ^• 

"^ a — b a — b a — b 

146. When the coefficient of the unknown term is a frac- 
tion, the value of the unknown number is found by multiplying 
both members of the equation by the denominator (Ax. 3), and 
dividing both resulting members by the numerator. Thus, if 

(1) faj = 15, 3a?=15 x 4 = 60, and aj = 60^3 = 20. 

2ab 
a 



(2) — aj = 2 a, oaj = 2 a&, and x = = 2 6. 



It is really unnecessary to refer to Axioms 3 and 4 in the above solu- 
tions : for if 3a; = 30, it is evident that ac = J of 30, or 30 -^ 3 = 10 ; and 
if Jaj = 15, it is evident that 3x = 16 x 4 = 60, and a; = 60 -f- 3 = 20. 
Indeed, the relations involved are as evident as the axioms cited ; and, 
besides, the pupil has applied these elementary principles throughout his 
entire course in arithmetic. 

The equations hitherto solved, and those to be solved in this chapter, 
are integral; that is, equations no one of whose terms is fractional 
(§ 129). The solution of fractional equations will be presented in 
Chapter X. 

Transpose the terms and find the value of x in 

3. 3aj-12 = 18. 9. 3a;-8 = 16-5«. 

4. 6«-5 = 3aj + 10. 10. 12 aj - 64 = 4(aj - 5). 

5. 5a? + 15 = « + 35. 11. 2(3aj-4)=3a? + 13. 

6. 52-30?= 73 -10a?. 12. 5a?-(2a?+15)=27-4a;. 

7. 12a;-5 = 9aj4- 13. 13. 2a? + 3 = 16 -(2aj- 3). 

8. 3a;-3 = a; + 5. 14. 5 a? -12=38 -(2 a; -13). 



78 ALGEBRA. [§ 146. 

Find the value of x in 

16. 8aj-(a?-12)=3aj + 16. 

16. 4aj-25 = 30-(2iB-5). 

17. 6x-6(x-5)=7(x-S)-'5. 

18. a? -[3 +(« - 3 + a?)-t- 2] = 7. 

19. 6aj-(12-a?)=24-(3aj-4). 

20. 2a?— (4a; — l)=5a?— (a?-hl). 

21. 6a?-3(a?-l)=32-(5 + 2a?). 

22. 5a? + 5 4-6(aj + 2)= 9(aj + 3). 

23. 3 aj -(3a? - 3 4- 2 a;)=2(3 - ^aj). 

24. (a?-2)(a? + 3) = (a?-5)(a? + l)+24. 
26. (a?-3)(a? + l) = (a?-l)(a?-3). 

26. 7a? -(2a? + 10)= 40 4- 4(a?- 12). 

27. a? + 2(a?-5)=5(13-a?)-3. 

28. 2(a? + 3)- 5(a?- 7)= 3(a?- 14)- 1. 

29. 32(a?-5)=27(a?-3)-4. 

30. 2(a? + 3)=5(a?-l)-(-2a?-l). 

31. 5(a?4-l)-3(a?4-5)=2(6-a?) + 2. 

32. 2a?+(a — 6)=a + 6. 

33. ax-\-b = 2a + b. 

34. aa? 4- 6 = a + &a?. 

35. 3aa? + 6 = 2aa?— (a — 6). 

36. 4:X — 2a = 6a + 2x. 

37. aa? — (a — &)= 2 a + b. 

38. ax — ab = b^— bx, 

39. (aa? + 2)(a + l)=a(a? + 2)+5. 

40. (a4-a;)(a-a?)4-&=(2 4-a)(l-»). 

41. oa? — 6a?=(a + 6)(a — 6). 

42. a(a? — a)=6(a? — 6). 

43. 2(a?-&)+3(a?-2 6)=2&. 

44. {a'^b)x-\-{a—b)x^2a^b. 



§150.] SIMPLE EQUATIONS. 79 

147. To solve an integral equation of the first degree with 
one unknown number, 

Transpose all the terms containing the unknown number to 
the first member, and all the other terms to the second member. 

Combine the like terms in each member, and then divide both 
members by the coefficient of the unknown number. 

To verify the result, substitute the value of the unknown numr 
ber in the original equation. 

PROBLEMS. 

148. A problem is a question proposed for solution. 

149. The solution of a problem consists of three processes 
or steps : namely, 

I. The framing of algebraic expressions to denote the 
different numbers in the problem, called the notation. 

II. The expression of the given relations between the 
known and unknown numbers in the form of an equation, 
called the statement. 

III. The finding of the value of the unknown number, 
called the solution of the equation. 

The first two of the above processes are usually called the statement. 

It will be seen later that the statement of a problem may involye the 
forming of two or more related equations, and its solution the finding 
of the values of the imknown numbers in the several equations. 

150. Any given equation may be regarded as the algebraic 
statement of a problem; and the invention of problems for 
which a given equation is a statement, is an interesting and 
useful introduction to the inverse process. Thus, x-\-2x = 4t5 
may be regarded as the algebraic statement of the problem, 
"A's age is twice B's age, and the sum of their ages is 45 
years." The papil has now had sufficient practice in the 



80 ALGEBRA. [§ 150. 

notation and statement of problems to make such inventions 
of problems easy. 

Invent problems of which the following equations are state- 
ments : 

1. x + Sx = 48. 6. aj + 2a;-f8 = 44 

2. 5 a; — aj = 24. 7. a? + 24 = 6 a?. 

3. 3aj-fa; = 16. 8. 3aj + 4 = aj-f 20. 

4. 2a; -a? = 18. 9. 3a?-2aj = 3. 

5. a; + aj + 8 = 24. 10. 2 a; -f 3 a; = 400. 

11. Five times a certain number less 9 equals 15 less 3 
times the number. What is the number? 

Let X = the number ; 

then 6a;-9 = 15-3«. 

Transposing terms, 6a;4-3a; = 16 + 9; 

combining terms, 8 a; = 24 ; 

dividing by 8, x = 3. 

Verification. 6x3 — 9 = 15 — 3x3; 

that is, 16-9 = 15-9. 

12. If 3 times a number less 6 be subtracted from 5 times 
the number, the difference will equal 24 less 4 times the num- 
ber. What is the number ? 

Let X = the number ; 

then 6a; - (3a; - 6) = 24 - 4a;. 

Removing parenthesis, 5a; — 3a; + 6 = 24 — 4a;; 
transposing terms, 6x — 3« + 4a; = 24 — 6; 

combining terms, 6 x = 18 ; 

dividing by 6, x = 3. 

Verification. 6x3- (3x3- 6) =24 -4x3; 
that is, 15 - 9 + 6 = 24 - 12, or 12 = 12. 

13. Divide a line that is 25 inches long into two parts such 
that the greater shall be 4 inches longer than twice the less. 



§ 150.] SIMPLE EQUATIONS. 81 

Let X = length of less part ; 

then 2 X + 4 = length of longer part. 

Hence x -f 2 x + 4 = 25. 

Transposing terms, x-f2x = 25-4; 

combining terms, 3x = 21; 

dividing by 3, x = 7, less part ; 

2 X 7 + 4 = 18, longer part. 

14. A father's age is twice his son's age, and 10 years ago 
his age was 3 times his son's age. What is the present age 
of each ? 



Let 


X = son's present age ; 


then 


2 X = father's present age ; 




2 X — 10 = father's age 10 years ago ; 




X — 10 = son's age 10 years ago. 


Hence 


2 X - 10 = 3 (X - 10) = 3 X - 30. 


Transposing terms. 


2x-3x= -30 + 10; 


combining terms, 


- X = - 20 ; 


multiplying by — 1, 


X = 20, son's age ; 




2 X = 40, father's age. 



15. The difference between two numbers is 17, and their 
sum is 93. What are the numbers ? 

16. What number added to 7 gives a sum equal to twice 
that number increased by 1 ? 

17. The sum of two numbers is 54, and their difference 10. 
What are the numbers ? 

18. A man is 5 years younger than his brother, and tlie 
sum of their ages is 55 years. How old is each ? 

19. A father is 25 years old, and his son is 5 years old. In 
how many years will the father's age be twice the son's age ? 

20. A father's age is 4 times the son's age, and the differ- 
ence of their ages is 27 years. What is the age of each ? 

21. Divide $ 1000 among A, B, and C, so that A shall receive 
$ 72 more than B, and C $ 100 more than A. 

white's alo. — 6 



82 ALGEBRA. [§ 150. 

22. A man sold a horse and buggy for $ 200 ; and one third 
the price of the horse was equal to one half the price of the 
buggy. Find the price of each. 

Suggestion. Let Sx = price of horse, and 2x = price of buggy. 

23. Divide the number 72 into two parts such that 3 times 
the greater may be equal to 5 times the less. 

24. In a school of 143 pupils 5 times the number of boys 
equals 6 times the number of girls. How many of each ? 

25. I bought equal numbers of one-cent, two-cent, and four- 
cent stamps, paying $ 1.05 for all. How many did I obtain of 
each kind ? 

26. Divide the number 40 into three parts such that the 
second may be 3 times the first, and the third double the 
second. 

27. At an election the number of votes cast for both can- 
didates was 2560, and the successful candidate had a majority 
of 500 votes. How many votes did each receive ? 

28. Divide $2147 between A and B so that 8 times A's 
share may be equal to 11 times B's share. 

29. A farmer, being asked how many sheep he had, replied 
that, if he had 12 sheep more, he would have 100 less than 
double the number he had. How many had he ? 

30. After A had received $ 12 from B, he had $ 19 more 
than B, and between them they had $75. How much had 
each at first ? 

31. The sum of $82.50 was paid in dollars, half-dollars, 
dimes, and five-cent pieces, an equal number of each piece 
being used. How many pieces of each kind ? 

32. A man paid $ 1000 for a certain number of horses at 
$ 60 each, 3 times as many cows at $ 30 each, and 20 times as 
many sheep at $ 5 each. How many of each did he buy ? 

33. A bill of $ 34 is paid in half-dollars and dimes, just 100 
coins being used. How many of each were used ? 



§150.] SIMPLE EQUATIONS. 88 

34. A man divided $ 9 among a number of children, giving 
to some a quarter each, and to twice as many a dime each. 
How many children received the money ? 

d5. The sum of $ 15,000 was raised among A, B, and G ; B 
contributed $ 500 more than A ; and as much as A and B 
together. How much did each contribute ? 

36. Each of five brothiBrs is 3 years older than his next 
younger brother, and the oldest is twice as old as the youngest. 
What is the age of each ? 

37. Divide the number 18 into two parts such that 5 times 
the greater increased by 4 shall be equal to 9 times the less 
diminished by 4. 

38. Divide $ 4400 among A, B, and G, so that A may receive 
twice B's share, and B three times G's. 

39. A company of 90 persons consists of men, women, and 
children. There are 4 more men than women, and 10 more 
children than both men and women. How many of each in 
the company ? 

40. A boy engaged to carry 100 glass vessels to a certain 
place on the condition that he should receive 3 cents for each 
one carried safely, and pay 9 cents for each one he broke. On 
settlement he received $ 2.40. How many did he break ? 

41. The sum of f 5000 was divided among four persons, so 
that the first and second received together $2800; the first 
and third together, $ 2600 ; and the first and fourth together, 
$ 2200. How much did each receive ? 

42. After 34 gallons had been drawn from one of two equal 
casks, and 80 gallons from the other, one cask contained 3 
times as much as the other. How much did each contain at 
first? 

43. A brother is twice as old as his sister, and 3 years ago 
he was 3 times as old as she. What is the age of each? 



84 ALGEBRA. [§ 150. 

44. In a brigade of 4500 men, the cavalry was 50 less than 
twice the number of artillery, and the infantry was 200 more 
than 8 times the cavalry. How many were there of each arm ? 

45. The distance between two cities is 1083 miles. From 
each a train sets out towards the other at the same hour^ one 
at the rate of 35 miles an hour, and the other at the rate of 22 
miles an hour. In how many hours will they meet ? 

46. Divide the number 70 into two parts such that the first 
plus 10 will be equal to the second multiplied by 3. 

47. A and B engage in trade with equal capital. A gains 
$ 1600, and B loses f 1900, and A's capital is now 8 times B's. 
What was the capital of each at first ? 

48. A boy has 5 more marbles in his right pocket than in 
his left ; but, if he transfers 8 marbles from his left pocket to 
his right, he will then have 4 times as many marbles in his 
right pocket as in his left. How many marbles had he in each 
pocket at first ? 

49. A father gives to his four sons $ 2000, which they are 
so to divide that each elder son shall receive $ 50 more than 
his brother next younger. What is the share of each ? 

50. A newsboy has $ 6.50 in quarters, dimes, and five-cent 
pieces ; and he has 3 times as many dimes as quarters, and 5 
times as many five-cent pieces as dimes. How many pieces of 
each kind has he ? 

51. If a steamer sails 9 miles an hour downstream, and 5 
miles an hour upstream, how far can it go downstream and 
return again in 14 hours ? 

52. If a steamer sails downstream at the rate of a mile in 
5 minutes, and upstream at the rate of a mile in 7 minutes, 
how far downstream can it sail and return again in 1 hour ? 



§ 154.] FORMULAS. 86 



CHAPTER VI. 

FORMULAS. 

SPECIAL FORMS IN MULTIPLICATION AND DIVISION. 

151. An algebraic equation may be the expression of a 
general principle or a rule. 

Thus, the equation — ^ — | — ^^— = a is the algebraic expres- 
sion of the principle, tJie half of the sum of two numbers added 
to lialf their difference equals the greater number. 

152. The algebraic expression of a general principle or rule 
is called a formula. 

153. The following formulas found by multiplication or 
division are of special utility in abridging algebraic processes, 
and also as a basis of factoring. 

154. I. The square' of the sum of two numbers. 

(a -h by=(a + b)(a 4- &); and it is found by multiplying, that 
(a + b)(a + b)= a^ -f 2a6 -f b\ Hence 

(a-\-by=a^-\-2ab-{-b\ (1) 

Since a and b in (1) represent any two numbers, it follows 
that 

The square of the sum of two numbers is the square of the first, 
plus twice the product of the first multiplied by the second, plus the 
square of the second. 

Write by Formula (1) the square of 

1. ic-hy. 4. a; 4- 3. 7. 4ic-hy. 

2. a + aj. 6. 5 + a?. 8. 2ax-{-S. 

3. a? -1-6. 6. 3a 4-6. 9. 3a-f-56. 



86 ALGEBRA. [§ 155. 

156. II. The square of the difference of two numbers. 

(a — b)^ = (a — 6) (a — 6), and by multiplying it is found that 
(a - b){a - b)= a? - 2ab -f- W, Hence 

(a - by = a* - 2 a6 + b\ (2) 

Since a and b represent any two numbers, it follows from 
(2) that 

The square of the difference of two numbers is the square of 
the first f minus twice the product of the first multiplied by tlie 
second, plus the square of the second. 

Write by Formula (2) the square of 

10. x — y. 13. 5 — X. 16. x — 5y. 

11. x — a. 14. ci? — b\ 17. oaj — 4. 

12. » — 3. 15. 3a — 6. 18. 4a — 36. 

156. The foregoing principles are best fixed in the memory 
by means of their formulas, as follows : 

(a4-6)' = a' + 2a6 + 6l (1) 

(a-6)2 = a2-2a6-|-62. (2) 

157. These two formulas may be united in one by the use 
of the double sign ± , read " plus or minus," thus : 

{a±by = a^±2ab^b\ (3) 

If the upper sign (-|-) or the lower sign (— ) is used in the 
first member of the formula, the same sign must be used in 
the second member. 

158. If both terms of a binomial have the same sign ia-\-b 
or —a — b), all the terms of its square will be positive; but, 
if the two terms of a binomial have unlike signs (a — b or 
— a-\-b)j the second term of its square will be negative. 

Thus, (a 4- 6)' or (- a - bf = a^ -f 2 a6 + 2>' J and (a- bf or 
(-a + 6)* = a«-2a6-f 61 



§160.] FORMULAS. 87 

Write the square of 

19. a-\-3x. 23. a* — 26. 27. —oft — 20". 

20. « — 3y. 24. 2a -3b. 28. c^b — 2cx, 

21. 2m 4-1. 25. 3a^ — 2b\ 29. aihf + 25. 

22. 1— 2 m. 26. —ax + 2y, 30. 4ai^ — 2ajy. 

159. III. The product of the sum and the dijfference of two 
numbers. 

It is found by multiplying, that 

(a4.6)(a-6)=a«-y. (4) 

Hence 

The product of the sum and the difference of two numbers is 
the difference of their squares. 

Write the product of 

31. (x '\- y)(x — y). 35. (ax — 6)(aa; -f- 6). 

32. (aj + 4)(a?-4). 36. (1 + 3a^)(l -Soj^). 

33. (3a; + 3)(3a;-3). 37. (4a*» ~3y)(4a*a; + 33^). 

34. (2a + 36)(2a-36). 38. (5aW + l)(5a%2 - 1). 

160. IV. The square of any poljrnomial. 

The square of a trinomial may be readily found by Formula 
(1) or (2) by including two of its terms in a parenthesis, thus 
changing it to a binomial. 
Thus, since a4-64"C=(a + 6)H-c (Associative Law, § 82), 
(a 4- & 4- c)^=[(a+6)-hc]*=(a-f-6)'+2(a+6)cH-c2 
=^a^^-2ab + ly^-{-2ac-\'2bc + <? 
= a^-^b^ + <?-if2ab + 2ac^-2bc. 

39. Find the square of a — b — c. 

(o - & - c)2 = [(a - 6) - c]2 = (a - 6)2 - 2(a - 6)c + C^ 
= a2 - 2 a6 + 62 _ 2 ac + 2 6c + c2 

5= a^ + 62 + 0* - 2 a6 - 2ac + 2 6c. 



88 ALGEBRA. [§ 161. 

40. Find the square of a-f-ft-fc — d 

Since a + 6-}-c — d=(a + &) + (c — <Z), 
[(« + &) + (c - d)]2 =(a + 6)2 + 2(a + 6)(c -d) + (c-d)^ 

= o2 + 62 + c2 4- (P + 2a6 + 2ac - 2ad + 2 6c - 2 6d - 2c(f. 

161. An inspection of the foregoing equations shows that 
the square of a polynomial having three terms or four terms 
is made up of (1) the sum of the squares of each of its terms, and 
(2) twice the product of each term multiplied in su^icession by eax^h 
of the terms that follow it. 

It may be shown by multiplying, that the square of any 
given polynomial is the sum of the squares of its several terms 
and twice the product of each term multiplied in successum b^ 
each of the terms that follow it. 

Find the square of 

41. x-\-y — z. 47. X -\- y -}- z -\- V, 

42. X — y -^ z, 48. x — y — z — v, 

43. x + y — l. 49. a — 26 — c + 2d. 

44. x — y — z. 50. a + 6 — c — d. 

45. 2a-6 + c. 51. a^ -[- 11^ — (? — d^. 

46. a + 36 — c. 52. a2-f-62-2c + d. 

162. V. The product of two binomials with a common term. 

It may be found by multiplying, that 

(1) {x-\-a){x-\-h) = a?-\-ax-{-hx-\-ah = a?-\-{a-\-h)x-{-ah\ 

(2) {x-\-a){x—h) = x^-\-ax—bx—ab = x^-{-(a—b)x—ab; 

(3) (x—a)(x-{-b) = x^—ax-{-bx—ab — ar^-h(— a+6)a;— a6; 

(4) {x—a){x—b) = x^—ax—bx-^ab = a^-^{—a—b)x-^ab, 

163. It is seen from these formulas that the product of any 
two binomials with a common term is made up of (1) the square 
of the common term, (2) the product of the common term multi- 
plied by the algebraic sum of the two unlike terms, and (3) the 
product of the unlike terms. 



§ 165.] FORMULAS. 89 

164. By the aid of the foregoing formulas, the product of 
any two binomial factors with a common term may be written 
without multiplying, care being taken to observe that the 
coefficient of the second term of the product is the algebraic 
sum of the two unlike terms, and that the last term of the 
product is the product of the unlike terms. Thus, 

(x+4:)(x - 7)= a^- 3x - 28, and (x-4){x + 7) = a^+3x-2S. 

Find the product of 

53. (a? + 4)(a? -f 3). 61. (a? - 15)(a; - 9). 

54. (a? — 6)(a; — 4). 62. (x-a)(x — b). 
65. (a;H-8)(aj — 3). 63. (a - 6)(a -|- 11). 
56. (x - 9)(a: + 7). 64. (9-a;)(9-^). 

67. (a; + 3)(aj - 6). 66. (a;-2a)(a;-3). 

68. (a; -f 5)(aj — 13). 66. (» + 5 c)(aj - 3 c). 

69. (a? - 10)(a; - 6). 67. (a?- 4 6) (a;- 4). 
60. (aj-12)(aj + 9). 68. {x — ab)(X'\-2ab), 

165. VI. The product of any two binomials. 

It is found by multiplying x + a by y -{-b that the product 
of any two binomials is made ug of 

I. The product of the first terms of the binomials {xy). 
II. The algebraic sum of the cross-products of the binomial 
terms ; i.e., the product of the first term of each binomial mul- 
tiplied by the second term of the other (bx + ay). 
III. The product of the second terms (a6). 

Thus, (1), (2aj + 5)(x - 7)= 2a^ +[2a; x (- 7) + aj x 5]- 35 

= 2aj2-9aj-35. 
(2), (x - b)(Sx + 26)= 3a^ +(2 bx -3bx)'-2V 

= 3aj*-6aj-26«. 



90 ALGEBRA. [§ 166. 

Find the product of 

69. (3a?-5)(»-7). 77. (2a-36)(a + 6). 

70. (2x-7){3x'{-5). 78. (5a;- 4y)(a;-f 2y). 

71. (4 m — 6)(3m — 4). 79. (2 m + 3 mn)(m — mw). 

72. (3a;-l)(a;-3). 80. (3a- 2 6) (a -f- 6). 

73. (3aj + l)(aj-2). 81. (3x + b)(x-2b). 

74. (aj-7)(5aj-12). 82. (5y - a)(2y -2a). 

75. (2a? + y)(3a?-22/). 83. {2 x - y)(2 x -\- S y). 

76. (2-3a6)(3-2a6). 84. (3a + 56)(5a - 76). 

DIVISION BY BINOMIAL FACTORS. 

166. Since division is the inverse of multiplication, it 
follows from the foregoing formulas that 

a-|- 6 aH- 6 

(2) a'-2a& + 6'^^_^. (4. «'-(^ + <')« + ^=a-a 
a — 6 a — 6 

167. A trinomial that is the product of any two binomials 
may be divided by either binomial factor by inspection. 

Thus, «'-3a-28^^_ ^ a'-3a-28^ 
a-h4 a-7 

Divide by inspection 

1. a^-2xy + f hj x-y. 7. 9a*-12a6+46«by 3a-26. 

2. a^ + 2 a% 4- «>' by a* 4- &. 8. 4aj*-12«23/'+92^ by 2aj2-32/«. 

3. ic*- 6a; + 9 by a? -3. 9. a* + 5aj-24 by a;-3. 

4. 9a*4-6aZ> + &* by 3a + &. 10. a' -16a + 60 by a -6. 

5. 4aV+12aa;+9by2aaj+3. 11. a^ — 6aaj-f- 6a^ by a — 3a. 

6. 26»'-10«2/+i/^by 6a;- j/. 12. ar' + 2caj-16c* by a; + 6c 



§ 171.] FORMULAS. 91 

168. There are several classes of binomials which are divis- 
ible by a binomial factor, and usually the quotient can be 
written directly. The following cases are often of special 
value in factoring. 

169. I. Difference of two squares. 

Since (a -f b)(a — 6)= a* — b*, 

(1) _-_ = a-6; (2) - — ~=a-f ». 

Hence the difference of the squares of two numbers is divish 
ible by their sum or by their difference. 

Divide by inspection 

13. aj* — 2/* by oj — y. 18. 9aW — 6* by Sax — b. 

14. aJ*-2^ by aj* + 3/«. 19. 100 -9a%* by 10-3a6. 

15. aJ* - 16 by a^ - 4. 20. 1 - 9 a^ by 1 - 3a^. 

16. 9aj*-9 by 3a?H-3. 21. 25aV-l by 5aaj*-f 1. 

17. 4a*-96* by 2a-36. 22. 16aY-9 by 4ai*y-3. 

170. II. Sum and difference of two cubes. 
It is found by dividing, that 

t±^=a'^ab-\-b'; (1) 

^tll^:=^a'^ab-\-b^, (2) 

a — b 

171. Formula (1) shows that the sum of the cubes of two 
numbers is divisible by their sum, the quotient being the square 
of tJie first number J minus the product of the first multiplied by 
the second, plus the square of the second. 

Formula (2) shows that the difference of the cubes of two 
numbers is divisible by their difference, the quotient being 
the square of the first number, phis the product of the first multi- 
plied by the second, plus the square of the second. 



92 ALGEBRA. [§ 172. 

Divide by inspection 

23. a:^4-2/* by fic-f y. SO. Qih^ — 7^ by xy^z. 

24. 31^ — j^ hy x — y. 31. a^y^-f 2;* by xy + z. 

25. Sa^ + d^ by 2a + &. 32. 27 + «y by 3 + a?y. 

26. 8a3-6« by 2a-6. 33. ixfi - f by a^ - f. 

27. x'-^lh^ by a;-36. 34. aJ« + / by ar^ + j/^. 

28. l-8y»byl-2y. 36. 1 - 64 ar^y^ by 1 - 4 iry. 

29. 82/^-1 by 2y-l. 36. 27 aj^yS - 8 by 3 a^ - 2. 

172. III. Sum and difference of other like powers. 

It is found by dividing, that 



a-\-h 
a — b 



= a^-a^b-\-ab^-l^', (1) 

= a^ + a^b-hab^ + y'y (2) 



a + 6 ' ^ ^ 

?LZ1^ = a* + a^ft + a«6« + a6« + 6*. (4) 

a — b 

It is also found by division that a^ + V, o? + 6*, and so on, 
are each divisible by a + 6 ; and that cH — V, d? — l?<, and so 
on, are each divisible by a — 6. 

173. It will, however, be found by trial that a' + 6', a* + 6*, 
a^ -H 6*^, a^ + 6^ and so on, are not divisible by a + 6 or a — &, 
and also that o^ — b^, a* — b^, and so on, are not divisible by 
a-\-b. 

It can be proved that, if n is any positive integer, 

I. a" + b" is divisible by a -f b if n is odd, and by neither 
a + b nor a — b ifn is even. 

II. a" — b" IS divisible bySL — bifn is odd, and by both a + b 
and a — b 1/ n is even. 



§ 174.] FORMULAS. 98 

174. It is seen from Formulas (2) and (4) in § 172 that the 
terms of the quotient are all positive when the divisor is a— 6; 
and from (1) and (3), that the terms are alternately positive 
and negative when the divisor is a-\-b. It is also seen that 
the exponents of a in the quotient decrease, and those of b 
increase, from left to right. 

Divide by inspection 

37. a^ — ^hyx + y. 44. 1 — iC* by 1 + «. 

38. iB!* — ^ by a? — y. 45. 1 -h «^ by 1 -f a?. 

39. ar^-hy* by x-\-y. 46. 16 — aJ* by 2 — a;. 

40. ar^ — y* by aj — y. 47. a:^ — 64 by a? -f 2. 

41. af^ — r^hyx + y. 48. aJ* — 81 by x — 3, 

42. a^ — / by a — y. 49. 8a?* + l by 2aj-f 1. 

43. aj*-l by aj-1. 50. a^-\-27f by x + 3y. 

MISCBLLAITBOUS EXBRCISBS. 

Write the square of 

1. x + y. 13. c -h d 4- e. 

2. 2 a? — y. 14. a — 6-f-c. 

3. m-hl. 15. aj + y — 2. 

4. 1 — m. 16. 1 + c — d. 

5. 2 -3a. 17. 3a-6-|-t 

6. 2x-Sy. 18. a-26 + 1. 

7. 3aj2-.22^. 19. x-2y-\-3, 

8. a^-^2b. 20. a + 26 + 3c. 

9. 5 — be. 21. 5a — 6— c. 

10. 2 aaj + y*. 22. m-f7i + r-ha. 

11. db — c. 23. m — w -f r — «. 

12. 7 — 2aa?. 24. a-h& — c — 1. 



94 



ALGEBRA. 



[§ 174. 



Write the product of 

25. (a + y)(a;-y). 

26. (m* — n){m^ -h n). 

27. (2 3/« + l)(22/«-l). 

28. (l-h3aaj)(l — 3aic). 

29. (5-3a;)(5-3a;). 

30. (7a-f-5)(7a-f 5). 

31. (4a*-f 2 6)(4a2-2 6). 

32. (aj2 + yO(aj2-y2). 

33. (a^ - f)(a^ ^ f). 

34. (a + 7)(a — 4). 

35. (» — 3a)(aj — 4 a). 

36. (x -f 6)(a; — c). 



37. (x — m){x + n). 

38. (w — 5)(m + 12). 

39. (m + r)(m + s). 

40. (ar* - aj + l)(aj + 1). 

41. (aj« + aj+l)(a-l). 

42. (a^'\-a? + x + l){x-l), 

43. (a^ — aa 4- iB*)(a -f x). 

44. (ic* + aa + tt^(a; — a). 

45. (i»*-2aj + 4)(aj + 2). 

46. (aj2 + 3aj + 9)(a?-3). 

47. Qx^ — a^ -\- xf — y^(x -\- y), 

48. (a? + 0^ + 02^ + 2/^(0? -2^). 



Write the quotient of 



49. 


aJ« + 2/* 


ic + y 


50. 


a^-a? 


a — x 


51. 


o* — 2/* 


«— y 


52. 


a^-a!« 


a + jc 


53. 


a' -64 


a-4 


Rd 


a^-81 



a + 3 



55. 



56. 



57. 



58. 



59. 



60. 



aJ*-9 



0^ + 3 


1- 


a* 


1 + 


a« 


a*- 


-1 


a — 


1 


aj5- 


-f 


aj- 


-y 


Sa^-^f 


2x 


+ 2/ 


8a» 


-21W 



2a-36 



61. 



62. 



63. 



64. 



65. 



66. 



9aW-166* 



3aaj + 46* 


4mV-25 


2m*n + 5 


a2-5a-24 


a-8 


aj2 + 4a?-32 


aj + 8 


a* - 2 a*6^ + 6* 


a2-6« 


aJ*-2a:V+2/^ 



x»-y* 



§ 179.] FACTORING. 96 



CHAPTER VII. 
FACTOROrG. 

175. The factors of a number are the numbers which, multi- 
plied together, will produce the number (§ 22). 

Since a x 1 = a, every number is equal to the product of 
itself and 1, and hence the number itself and 1 are factors of 
every number. 

176. A number which has no integral factors except itself 
and 1 is a prime number (§ 22). 

A number which has two or more integral factors besides 
itself and 1 is a composite number. Hence 

177. Every composite number may be resolved into two or 
more integral factors besides itself and 1. 

In giving the fjwjtors of a number, the number itself and 1 are usually' 
omitted. A prime factor is a prime number. 

178. Factoring is the process of resolving a composite num- 
ber into its prime factors. 

Skill in factoring is of great utility in abridging algebraic work, and 
the acquisition of such skill requires much practice. 

Case I. 

179. Monomials. 

Since 12 = 3 x 2 x 2, and a?=:a xax a, the prime factors 
of 12 a^ are 3, 2, 2, a, a, a. 

In like manner any monomial may be resolved into its prime 
factors by factoring the coefficient, and taking the ba^e of each 
Uteral factor as many times as there are units in its ea^onent (§ 29). 



96 ALGEBRA. [§ 180. 

Resolve into prime factors 

1. 6a36'. 4. -ISmViaj'. 7. 35a%V. 

2. 10a*6a». 6. 49 a»cd«. 8. -81aV. 

3. -9aY. 6. -121iBy. »• 100 mVs. 

180. A number that is composed of tWo equal factors is a 
perfect square, and one of the two equal factors of such a 
number is its square root, 

181. Since (Sd'f = 3a« x Sa^ = 3« x a»^« = 9a«, 

V9a? = V9 X Va« = 3 x a*^« = 3 a» 
Hence 

The square root of a monomial is found by extracting the 

square root of the coefficient, and dividing the exponent of each 

letter by 2. 

10. Resolve 16a*b^ into two equal factors. 

Vl6a*6« = 4a26; .-. 16 a*b^ = 4 a^b x 4 a^b. 

Vl6 a*b^ = ±ia^b (§ 336), but in this chapter only the positive square 
root is considered. 

Resolve into two equal factors 

11. 25 aV. 13. 121 mVr«. 15. 64a*6V. 

12. Slod'f, 14. 49mVs»». 16. lUa^s^. 

182. In like manner the cube root of a monomial is found 
by extracting the cube root of the coeffixiient, and dividing the 
exponent of each letter by 3, 

Thus, -^1250^? = -^^25 x a^^b^-^a^^ = 5 a%:x?. 

Case II. 

183. Pol3nioinial8 whose terms have a common monomial 
factor. 

This case presents no difficulty requiring explanation. 



§ 184.] FACTORING. 97 

1. What are the factors of 3aa:» H- 6a% - 3a«c? 

It is seen by inspection that 3 a is a factor of each term^ 
and hence, dividing by 3 a and writing the quotient in a paren- 
thesis with 3 a as the other factor, we have 

Sax^-h^ a^by - 3 a»c = 3 a (aj» -f 2 a5y - a*c) , 

2. Resolve into factors a^ -f- a^V ■+■ ^• 

««y + «*y^ + xy« = xy(x2 + xy + !^). 

Resolve into factors 

3. 10a% + 5a6». 8. 9 a*6a; - 15 a% - 3 a»6. 

4. Ua^ -6Sxy^, 9. 15 a^y - 10 a^ '\' 5 a^, 

5. aa^ - a^a^y + axf, 10. 4 a?^ - 12 a?y - 16 a?y + 8 aV- 

6. a*6 - a«6* + a«6». 11. 3 ary - 6 «V + 9 aj^y* - 12 ary. 

7. cf^ + ahj-a^ 12. af - 3 dffz + 6 a^^ - aS/^, 

Case III. 

184. Pol3rnomial8 whose terms grouped have a common factor. 
A polynomial of four terms can sometimes be so arranged 
that the first two terms and the last two terms have a common 
binomial factor. 

Thus, oa? + ay 4- 6a; -(- 6y = (ax + hx) 4- (ay -\-by) = (a-\'h)x 
-\-{a + h)y\ and, dividing (a4-6)a; + (a + 6)y by a + &, we 
have x-\-y\ and hence (a + 6) a; + (a -f- 6) y = (a -f h){x -{- y). 

1. Resolve into factors oar* — ay -{-by — ha?, 

ox^ — ay -\-hy — hx^ = (ox* _ jx^) — (ay — 6y 
= (a - 6) x2 - (a - 6)y = (a - 6)(x2 - y). 

The inclosing of the two terms, —ay-{- by, in a parenthesis 
preceded by the sign — , involves the changing of the signs 
(§ 104). 

2. Resolve into factors 2 aa? — 4 ay — 3 6a; -f- 6 6y. 

2ax-4ay-36x + 66y = (2ax-8 6x) - (^ay-Qhy) 
= (2a-36)x- (2a-3 6)2y=(2a-36)(x-2y). 
white's alo. — 7 



gg ALGEBRA. [§ 185. 

Resolve into factors 

3. a6H-6y + aa;-|-ajy. 8. s^ + y'-fy + l. 

4. a^ -\- ax — ay — xy. 9. a' — 3 6 — a^6 + 3 a. 

5. ay — ab — bx -{- xy. 10. aa^y + aby^ — a^a^ — a]^. 

6. asy — 2 my 4- 2 mn — naj. 11. 6 a' + 4 a* — 9 a — 6. 

7. ax" - af -\- bx" - bf. 12. 6a«a^-4ay-36a;»+262r'. 

Case IV. 
185. Trinomials which are perfect squares. 



Since (a-\-by=a^+2ab-{-b'(^ 154), Va^+2a6+y =a-F6; 
and since (a-6)2=a2-2a6+62(§ 155), Vc^^^2a6+^=a-6. 

186. It is thus seen that a trinomial is a perfect square, if, 
when arranged according to the powers of some letter, its first 
and last terms are perfect squares and positive, and its sec- 
ond term is twice the product of their square roots. Thus, 
9a^-\-6xy -{-y^ is s, perfect square. Hence 

The square root of a trinomial which is a perfect square is 
found by connecting the square roots of the terms which are 
squares with the sign of the remaining term. 



Thus, ^9x'-12xy + 4:f=V9x^-V^=Sx-2y, 

187. A trinomial which is a perfect square may be resolved 
into two binomial factors by extracting its square root, and mak- 
ing the result one of its two equal factors, 

1. Resolve into factors 4 aj* — 12 aj^ + 9 y*. 

V4^ = 2x; V9y* = 3y2; 2 a; - 3 y2 = the square root. 

Resolve into factors 

2. 4ar*+4iry4-2/'. 6. 9 a^ — 6 xy + y', 

3. a«4-6a5+96». 6. 4 aj* - 20 ar^t/* + 25 y*. 

4. a*-2a«6* + &*. 7. a^-2aj + l. 



§ 189.] FACTORING. 99 

8. Af-^y + 1. 14. 4-40a& + 100aV. 

9. 9a^'-24:xy + 16f. 15. a^^^ - 6 ajy* + 9. 

10. 25a^-10xy^-\-^. 16. 16 + 40 oft V + 25 a% V. 

11. 144 aj*- 120 arV+ 25 y*. 17. 121 a*+2200a6*+ 1000061 

12. 1-10 ay -1-25 ay. 18. m V - 40 m W + 400 «*. 

13. 25 + 30a6 + 9a%2. 19. (a + 6)2- 4 (a -|- 6) -|- 4. 

20. (aj-2/)2_6(a?-y)4-9. 

21. (7n!'-ny-10(m^-n)n + 25n\ 

22. a^m -|- a^n + b^m + 6^ — 2 a6n — 2 oftm. 

Arran^ng terms, (a^w — 2 a6m + b^m) + (a^n — 2 abn + b^) 

= (a2 - 2 aft + 62) ^ -\- (^a^ - 2 ab + b^)n 
= (a - 6)2w + (a - 6)2» = (a - 6)2(w -h n) 
= (a — 6) (a — 6) (m + w). 

23. a^ic* ^ 2 a^xy -f- a^y^ _i,2^_2 h'^xy - by. 

188. Since the square of a polynomial is made up of the 
sum of the squares of its several terms and twice the product 
of each term multiplied in succession by all the terms that 
follow it (§ 161), the square root of a polynomial which is a 
perfect square may be found by connecting the square roots of 
the several terms which are the squares with the proper signs, 

189. The signs of the terms of the root are determined from 
the several terms of the polynomial which are the products, 
and this can usually be done with little difficulty by inspec- 
tion. 

Thus, a^-2bc + 2ab + <^-2ac-^b^ 

=^a2 -h 6^ -f- c2 -h 2a6 - 2ac- 26c = (a + 6 - c)*. 

It is seen that each product containing c, viz., — 2ac, —2 be, 
is negative, and so it is inferred that c has the sign — . 

Resolve into two equal factors 

24. a:^ + 2xy-{-2xz-\-f-\-2yz-^s?. 

25. a^-2xy-^z^ — 2yZ'\'2xz-{-f. 

26. a*-4a26 4.4a2-t.462_86-j.4. 



100 ALGEBRA. [§ 190. 

Case V. 

190. Binomials expressing the difference of two squares. 

Since (a -\-b)(a-h)=a^ - h^ (§ 159), 

a»-6*=(a + ^)(a-6). 

It is thus seen that the difference of two perfect squares is 
equal to the product of the sum and the difference of their 
square roots. Hence 

191. A binomial expressing the difference of two perfect 
squares is resolved into two factors by extracting the square 
root of each of its terms, and then taking the sum of their roots 
for one factor, and the difference of their roots for the other 
factor, 

1. Resolve into factors 9ajV — 4:a% 



.-. 90*2/2 _ 4 a252 _ (3 a.2y + 2o6)(3a;2y - 2a6). 

2. Resolve into factors 3 gwj* — 12 ay^. 

3ax*--12a2/2_3flf(a^_4y2). Vic*_a.a. V4p = 2y. 

/. 3aic*--12ay2_3flj(aj2 4.2 2/)(x2_2 2/). 

3. Resolve into factors (m + n)^ — (m — rif, 

V(m + w)2 = m + w ; V(m — n)2 = m — n. 



.'. (w + w)2 — (m — w)2 =(m + w + w — n)(w + n — wi — n) 
= (w + TO + w — n)(w + « — m + «)=2wx2« = 4 mn. 

Resolve into factors 

4. 9a*62_l6c*. 11. 2^-162*. 

5. ^f-^.fz'. 12. 121 -aV. 

6. iB* — 2^. 13. iB* — 492/V. 

7. 9 2/^-1. 14. 144-25ay. 

8. 4aW-36al 15. a^-(a-6)*. 

9. 16aV-492*. 16. a^-{x-y)\ 
10. 1-812*. 17. 4aJ*-(a4-6)^ 



§ 192.] FACTORING. 101 

18. (x + yy-(x^yy, 22. (a + 6)* - (a - 6)*. 

19. (a + 6)*-(c + d)^ 23. (2a + 3)^ -(3a-4)«. 

20. (m-7i)2-(m + n)*. 24. (3a + 2 6)* -(2a- 3 6)«. 

21. (a-5)«-(c-(f)l 26. (5 a? + 2 y)* - (a? - 3 2^)1 

192. Some polynomials of four or more terms can be so 
written as to express the difference of two perfect squares, 
and their factors can thus be readily found. 

26. Resolve into factors ix^-{-2xy-{-y^ — n^. 

a;2 4- 2a;y + y2 _ ^ = (a;2 4. 2xy-{- t/^)-s^ 
= (x + yy-z^=(x + y + z)(x -{-y-z). 

27 . Resolve into factors 4a^ — a?* + 4aj^ — 42/*. 

4a2 - a* + 4a;3y - 43/a _ 4^2 _ (jc* _ 4x2y 4. 4y2) 

= 4a2 -(x2 - 2y)2 =(2a 4- «« - 2y)(2a - x^ + 2y). 

Special care must be taken to change the signs of terms when put in 
a parenthesis preceded by the sign — , and also when a parenthesis pre- 
ceded by the sign — is removed. Thus, above, — ac*-f-4a;22/ — 4j/2 be- 
comes — («* — 4 aj2y -I- 4 ^) J and — (x^ — 2 y) becomes —x^-\-2y. 

28. Resolve into factors a^ -^y^ — a^ — b^ — 2icy + 2ab, 

x^ + t/^ - a^ - b^ - 2xy + 2ab = x^ - 2xy + 1/^ -(a^ - 2ab + b^) 
= (25 - yy -la - 6)2 =(aj -y + a- b)(x - y - a + &). 

Resolve into factors 

29. 4a*-4a*6-ajy + 6^ 

30. 4a?*— 9ar^-f 6a;-l. 

31. a^b^ + c'-2abc-25aV. 

32. a^-4:cx?f-2ab + b\ 

33. l-2a? + aj2-16a%*. 

34. a2 + 46*-4a6-4ar*-92/^-12a:2/. 

35. a^-52-c2 + 2a + 26c + l. 

36. a?-{-y^ — 2xy — 2 mn — m^ — n*. 

37. i)^ + f-8i^-z^ + 2xy + 28Z. 
88. a^-2ac-62_^2_2M + c2. 



102 ALGEBRA. [§ 193. 

193. A polynomial of the general form a* -|- a%^ + 6* may 
be written as the difference of two squares by adding to it a^^, 
and then subtracting a%* (the number added) from the result. 

Thus, a* + a%^ + 6* = a* + a%* 4- 2>* -f a^ft^ _ a^ft* 

= (a^ -f by - a^h^= (a" 4- &' + a6)(a* + 6^ - a6). 

39. Resolve into factors a^ + a^ + 1. 

= (a^ + 2x2+l)-x2=(a;2+l)2-a;a 

40. Resolve into factors 9 a* — 3 a%* + 6*. 

9a* - 3a2&2 + ft* = 9a* - 3a262 + 6* + 9a2&2 - 9a26» 
= 9a* + 6a262 + 54 _ 9^2^-2 =(3^2 + 52)2 _ 902^2 

= (3a2 + 3a& + 62)(3a2 - 3a6 + 62). 

Resolve into factors 

41. aj* -f ar^i/2 _|_ y4 44 ot^ + a^ft^ + ft*. 

42. aj* + 2a^2^ + 92^*. 45. a:^ + 2 a^ + 9 3/*. 

43. m*-8mW + 4w*. 46. 4 a;* - 16 a^^ + 9 2^. 

194. In like manner any polynomial that can be made a 
perfect square by adding to it a number, can be factored by 
adding such number, and subtracting it from the result, and 
then proceeding as in § 192. 

Thus, a* 4- 4 6*= (a* + 4 aV-{-4: b") -A:aV= (a^-{'2 b^- 4 d'b^ 

= {a^-{-2b^ + 2ab)(a^ + 2b^-2ab). 

47. Resolve into factors a^ 4- 41/*. 

a;8 + 4 y* = a;8 + 4 o^y- + 4 y* - 4 x^y^ 
= (rc* 4- 2y2)2 _ 4a;V =(«* + 2y2 •^2xhf)^a^ + 2y2 _ 2a;2y). 

195. If the number added and subtracted be not a perfect 
square, the resulting factors will contain a radical, or indicated 
root (§ 370). 

Thus, a^ ^ ab + V^ :=a^ -{- ab + b^ f db — ab 

= (a -{- by — db ={a -i- b 4- Va^)(a 4- & — Vab). 



§ 199.] FACTORING. 108 

Kesolve into factors 

48. a?* + 4. 50. a?-\-ix, 52. a* + 3 a'6* + 4 6*. 

49. 64 + 2^. 61. aj* + 2icy. 63. a*-6a%* + 6*. 

Case VI. 

196. Trinomials having^ binomial factors with a common term. 

Since by § 162 (a? + d){x + b) = a^ + (a -{- b)x + ab, 
conversely ic* + (a + b)x -\- a^ = (x -{- a)(x + b). 

Hence a trinomial of the form a^ + mx -\- p may be resolved 
into two binomial factors, if its third term is the product of two 
fcLctors whose algebraic sum is the coefficient of the second term, 

197. There are two cases : 

I. When the final term is positive. 
II. When the final term is negative, 

198. I. When the final term is positive. 
By § 163, we have, conversely, 

a^ + (« H- b)x + aZ> = (a? + a)(aj -h 6) ; 
«* — (a + b)x + ab ={x — d)(x — b), 

199. It is seen from these formulas, that, if the third term 
of the given trinomial is positive, the second terms of its 
binomial factors will have the same sign as the middle term of 
the trinomial ; and (2) that the sum of the second terms of the 
binomial factors is the coefficient of the middle term of the 
trinomial. 

1. Resolve into factors aj* + 7ic + 12. 

12=3x4;3 + 4 = 7. 

.'. x^+7x+ 12 =(« + 3)(a; + 4). 

2. Resolve into factors aj^ — 12 a; + 35. 

35 = - 7 X (- 5) ; _ 7 + (- 5) = - 12. 
.*. »2 _ i2x + 35 =(x - 7)(x - 5). 



104 ALGEBRA. [§ 200. 

Resolve into factors 

3. aj* 4- 7a; 4- 10. 6. 3^4.93.^20. 

4. ic«-13aj + 22. 7. JC* + 15a; + 66. 

5. a«- 14a? 4- 45. 8. aj^ - 11 a? -f 28. 

200. II. When the final term is negative. 
By § 162, conversely, we have 

a;* -f(a — b)x — ab=(x-\- a)(x — b) ; 
»* + (— a + b)x — ah=(x — a)(x 4- 6). 

201. It is seen from these equations that when the final 
term of a trinomial is negative, the second terms of its two 
binomial factors have unlike signs, and (2) that the algebraic 
sum of the second terms of the binomial factors is the coefficient 
of the middle term of the trinomial. 

9. Resolve into factors a;^ + 3 a? — 28. 

-28=7 x(-4); 7-4=3. 
.-. a;2 + 3x - 28 =(a + 7)(x - 3). 
The third term, — 28, also equals — 7x4; but — 7 -f- 4 = — 3, whereas 
the coefficient of the second term of the trinomial is + 3. 

10. Resolve into factors ar^ — 4 a; — 45. 

-46 = -9x6; -9 + 5=- 4. 
/. x2 - 4x - 46 =(x - 9)(a + 6). 

It is seen from Examples 9 and 10 that the sign of the 
middle term of the trinomial is the sign of the numerically 
greater second term of the binomial factors. 

Resolve into factors 

11. ar^ + 5aj-24. 15. a^ + 10a?-56. 

12. a^-7a?-60. 16. ar^-3aj~70. 

13. a2-4aj — 45. 17. a^ — X'-72, 

14. a^H- 2a; -63. 18. d^-^x-42. 



202.] FACTORING. 106 

202. It is thus seen that a trinomial whose binomial factors 
have only one common term may be resolved into its factors 
(1) by taking the sqvxire root of the first term for the first (yr comr 
mon term of the two factorSy and then (2) finding for their second 
terms two numbers such thcU their product is the final term of the 
trinomial, and their algebraic sum the coefficient of its middle 
term. 

Resolve into factors 

19. a^ + 8a; + 16. 38. z^-'Z-272. 

20. a^H- 14a; + 40. 39. a^ + 32 a; -f 176. 

21. a;2 + 16a;4-63. 40. a?-fl0aJ-76. 

22. a^ + 18aj + 72. 41. aj* - 40 a? -f 400. 

23. 2^ + 20y-f96. 42. aj*-faj-166. 

24. a^-15aa; + 64al 43. f^Vly-{-m. 

25. aj2-8aaj + 16a*. 44. aj* - 23 a; + 76. 

26. aj2-16aj + 66. 45. aj2-15a;-54. 

27. aj*-17aj + 62. 46. f^lly-42, 

28. 3^-272^ + 140. 47. t^-7y-170. 

29. a^ + 3a;-28. 48. f + 24.y + l^^. 

30. aj« + 7aj-18. 49. a^ + 17 aj« + 66. 

31. a^ + 6aj-66. 60. a^-15a:8 + 56. 

32. a:«H-a:-132. 51. aj8 + 2a^-99. 

33. 2r^ + y-182. 52. f + lf-S. 

34. xV-9a^-22. 53. 2^«» - 19 y' -f 48. 

35. a^3/* - 5 a^ - 104. 54. 7? + (p -^ c) x -\- be, 

36. a* — 7aa; — 60al 55. oi? -\- (a—c)x — ac. 



87. aj«-2a:-99. 56. y" - (a? ^l^f ^a^b 



2 



106 ALGEBRA. [§ 203. 

203. The required factors of the final term can usually be 
found by inspection; but when the term is large, or contains 
many factors, it may be advantageous to resolve 

it into successive sets of factors, beginning with 270 = 2 x 135 

small prime divisors as first factors, and then 3 x 90 

taking their multiples, as is shown at the right. 6 x 54 

6 x45 

57. Eesolve into factors a^ + 33 a? + 270. 9 x 30 

270 = 16 X 18 ; 16 + 18 = 33. 10 x 27 

.-. x^ + 33a; + 270 = (x + 16) (x + 18). 15 x 18 

Resolve into factors 

58. aj«-aj-240. 63. a^-aj-420. 

59. a^ + 31 a? -h 240. 64. aP + x-1260. 

60. 2^ + 102^-299. 65. a^ + 38a? + 240. 

61. 2^2-142^-480. 66. a^ -f 7 a; - 1320. 

62. «*-47aj + 540. 67. a^ + a; — 552. 

Case VII. 

204. Trinomials having^ binomial factors with unlike terms. 

We have considered the factoring of trinomials when their 
binomial factors have like terms, as (a ± 6)^, and alscT when 
their binomial factors have a common term, as (a? ±a)(x ± 6). 

In the remaining case the binomial factors have unlike terms, 
and the trinomial has the form of aa^ -{-bx-{-c. 

205. It follows from § 165, that, when a trinomial is the 
product of two binomial factors with unlike terms, 

I. The first term of the trinomial is the product of the first 
terms of its binomial factors. 

II. The third term of the trinomial is the product of the 
second terms of its binomial factors. 

III. If the third term of the trinomial is positive, the second 
terms of its binomial factors will both have the sign of its 
middle term. 



§ 205.] FACTORTNG. 107 

rV. If the third term of the trinomial is negative, the 
second terms of its binomial factors will have unlike signs. 

V. The middle term of the trinomial is the algebraic sum 
of the cross-products of the terms of its binomial factors. 

ft 

1. Resolve into factors 6 a?^ — 19 a; -f 10. 

6x^ = Sxx2x; 10=-2x(-6); 3 a; x (- 2)+ 2 a; (- 5) = -19a;. 
.-. 6a;2 - 19a; + 10 =(3a; - 2)(2a; - 5). 

Since the second term of the trinomial is negative, and the 
third term positive, the second term of each binomial factor is 
negative. 

6a^ = 6 xx^ or 6xx xoT Sx X 2a;, and 10 = 2 x 5 or 1 x 10. 
It is seen that 1 and 10 are not the factors to be taken, and that 
3 X and 2 x are the only factors of 6 a;^ which by cross-products 
with 2 and 5 will give — 19 x, the second term of the trinomial. 

2. Resolve into factors 4:0? -\-4:ajy — 3y^. 

4x^ = 2xx2x; -Sy^ = Sy x(- y) ; 2x x Sy + 2x x(-y)=4xy. 
.'. 4x^-\-4xy-Sy^ = (2x + Sy)(2x-y). . 

Since the algebraic sum of the cross-products must be 
positive, the greater product must be positive ; and hence 
3^ is +, and y is — . 

3. Resolve into factors 6 oc^ — ocy — 35 t^. 

6x^ = Sxx2x; - S^y^ = - ^y x 7 y ; Sx x(- by) + 2x x7 y = - xy. 
.'. 67^ -xy- 352/2 =(2a; - 5y)(3a; + 7 y). 

Resolve into factors 

4. 3ar^-22a?4- 35. 11. 6a?-xi/-2if. 

5. 6a;2-lla?-35. 12. 5a? -[-6xy -Sf. 

6. 12 m^- 31m + 20. 13. 2 rn? + m?n - 3 w?n\ 

7. 3a?-10x-{-3. 14. 3a^ + ab-2b\ 

8. 3a?^5x-2. 15. 10f-12ay-h2a\ 

9. 5ar^-47a; + 84. 16. 4.0? -\-4:xy -3f. 
10. 3o?'-5bx-'2l^. 17. 15ar^-4a^-352r*. 



108 ALGEBRA. [§ 206. 

206. A trinomial of the form aaf-i- bx + c can be readily 
factored by Case VI. if it be first changed into an equivalent 
trinomial having unity for the coefficient of its first term. 

18. Resolve into factors 3 oj^ — 31 a; + 66. 
Multiplying by 3, and dividing the result by 3, we have 

It is seen that 3 x is treated as x in Case YI. 

19. Resolve into factors 12 aj* + 47 a? + 45. 

= (i^^±25^P^±i!)=(3« + 5)(4x + 9). 

Since 12 =4 x 3, the first factor, (12x + 20), is divided by 4; and 
the second factor, (12 a; + 27), by 3. 

Resolve into factors 

20. 3ar* + 14a; + 15. 26. 6a^-aj-40. 

21. 7ar^ + 20aj + 12. 26. 6a^-13a?4-6. 

22. 5ar^-17a; + 12. 27. 3a^ + 5aj-12. 

23. 6a^-\-5x-e. 28. 3 ar' - 17 a? + 20. 

24. Sa^'-x -10. 29. 10 aj^ + 23 a? - 21. 

GENERAL METHOD OF FACTORING TRINOMIALS. 

Note. The study of this method may be omitted until the pupil 
reaches quadratic equations (Chapter XV.). 

207. Any trinomial of the form x^ + bx + c may be factored 
by changing it to an equivalent expression of the difference of 
two squares. 

Thus, whatever may be the value of 6, a^ + bx is made a 



§ 208.] FACTORING. 109 

perfect square by the addition of ( - j , for a* -f- 6a; -h [ - j 

==[« + -] (§ 186). Hence, by adding and subtracting ( „ ) ^ 
fic* H- 6a? + c, we have 

"■^-HD"-a)'-=(-i)"-?- 

It is evident that if 6* — 4 c is a perfect square, both factors 
will be rational. 

1. Find the factors of a* + 6 a; + 8. 

a;2 + 6x + 8 =(x2 + 6x + 9)- 9 + 8 =(x + 3)2 - 1 
= (x + 3 - l)(x + 3 + l) = (x + 2)(x + 4). 

2. Find the factors of x^ -h 9 a; + 20. 

3. Find the factors of 3 aj* — 5 aj — 2. 

3x2-5x-2 = 3(x2-fx-i) = 3[xa-ix+({)a-ff-f] 

= 3[(x - i)3 - |S]= 3(x - f + J)(x -i- J) = 3(x + i)(x-2). 

4. Find the factors of 2 a?* — 3 maj — 2 m*. 

2 x^ - 3 wx - 2 w2 = 2^x2 -l^x - mA 

o/«. 3m , 6w\/ 3m 6m\ «/ . i n^ ox 
= 2^x-— + — j^x- — -— j = 2(x + im)(x-2m). 

208. In like manner any trinomial of the form aoi^ -{-bx-\-c 
may be factored. For 

\ a aj \_ a \2oy 4 a' aj 
_ / & Vy34i^Y 6 Vy-4ac\ 



110 ALGEBRA. [§ 209. 

This is a general formula for all of the more important cases 
of factoring, and the preceding methods of factoring trinomials 
may be considered as special cases under it. It may be used with 
advantage in the solution of certain quadratic equations (§ 451). 

209. Any trinomial may be directly factored by substituting 
for a, h, and c, in the formula in § 208, their values in the 
given trinomial. 

Thus, in 3aj2 - lOa + 3, a = 3, 6 = - 10, and c = 3. Sub- 
stituting these values in the formula, we have 



, h , V62-4ac , -10 , VIOO - 36 1 



, b -Vb^ - 4 ac , -10 8 « 

x-{-- = x-\ = a? — 3. 

2a 2a 6 6 

Hence 3a^-10a? + 3 = 3(aj- ^)(x - 3). 

Find by general formula the factors of 
6. 3ar^-h5aj-12. 8. 3a^-13ic + 14. 

6. aj2 + 7aj-fl2. 9. 2x^ — 5mx — Sm\ 

7. 2a^ + x-2S. 10. 3x^-5nx-2n\ 

SUM OR DIPFERENCE OF LIKE POWERS. 

210. It follows from § 171, and also from Case V., that the 
difference of the even powers of two numbers can be resolved 
into two or more binomial factors. Thus, 

a^ - 52 = (a + b) (a - 6) ; 

a^-b*= (a' + b^ (a^ - b") = (a^ + 6^) (a + b) (a-b); 
a^-b^ = (a« + b^ (a« - b^ (for factors of a^ - b^ see § 212) ; 
a« - b'=(a* + b') (a^ - b') = (a*+b*)(a^-{-b^(aA-b)(a-b)] 
and so on. 

211. It has been shown in § 173 that the sum of the even 
powers of two numbers is not divisible by their sum or difference, 
and it will be found by trial that the sum of the even powers 



§213.] FACTORING. Ill 

of two numbers is not divisible by the sum or difference of 
any powers of the numbers except when the eocponents of such 
even powers contain an odd factor. 

Thus, a* + b^j a* -f- b\ a^ -f b% and so on, are not divisible by 
the sum or difference of any powers of a and b ; but a* -j- 6* 
= (ay + (by, and a^^ + &'^ = («")'+ (py, and are each divisible 
by a^ + b^y and hence can be factored. Thus, 

212. It follows, conversely, from §§ 170, 172, that 

a^ + b^=(a-{- b) (a^ -ab +b^; 

a'* - 6' = (a - b) (a^ + 06 -|- 6^ ; 

a? + «»* = (a + &) (a* - a^?^ + a%2 _ a5« + 5^) . 

a^ _ 6* = (a - 6) (a* + a% + a^^ -h a6« + 6*); 

and generally that the sum of any two like odd powers of a 
and b may be factored by dividing such sum by a + 6 ; and the 
difference of any two like odd powers of a and 6, by dividing 
such difference by a — b. 

213. In like manner many binomials whose terms have 
unequal exponents may be factored provided these exponents 
have a common factor. 

Thus, aj^°— 2/"=(a^*— (2/^*, hence has the factor a^ — T^; and 
m* — n^^ = (my — (ny or (m^y — (ny, and hence has as a 
factor either m* — n^, or m* + n^, or m^° — n^, or m^^ + n^, 

1. Resolve into factors (1) o^ + t/^; (2) a^ — ^. 

(1) a;8 4- 2/3 =(« + 2/)(«2 - ajy + y^). 

(2) ix^-yi=(^x- y)(«2 + ay + yS). 

2. Resolve into factors afi — 1/^. 

(xfi — y^ = (x^ -{- y^) (x8 — y8). 

a^ + y' = (x + y)(x2 - «y + y^) ; 
aj8 _ ys _ (a; _ y)(a;2 4. xy + y2). 

.-. x^ - y« = (x + y)(a; - y) (x^^xy + y^)(x^ + ay + y®). 



112 ALGEBRA. [§ 214. 

S. Resolve into factors 8 a«&» - 125 A 

v^So^P = 2 a6 ; \/l26c« = 5ca. 
. •. (dividing 8 o'd* - 126 c« by 2 aft - 6 c?) 

8a«6« - 126c« =(2a6 - 5c2)(4a«6a + 10a6c2 + 26c*). 

4. Resolve into factors (a — 6)' — 8 c*. 

^ v^(a-6 )« = a -5; v^8^ = 2c. 

.'. (dividing a — b - 8 c« by a — 6— 2c) 

(a _ 6)8 _ 8c« = (a - 6 - 2c)[(a - 5)a + (a - 6)2c + 4c2J. 

Resolve into factors 

5. of* + 8. 14. m^-n\ 

6. a^ — 1. 15. m*4-w'. 

7. l+iB«. 16. ar^-1. 

8. aj8-27. 17. l-\-(a + by. 

9. 125-|-iB«. 18. (a -6)8 + 1. 

10. 16 — a?*. 19. (m + w)* — (m — n)*. 

11. iC*-256. 20. (m-f w)«-(m — n)'. 

12. a®-27. 21. (2a-ha;)*-(a + 2a;)\ 

13. a^-2/«. 22. Sa^-(a-\-2xy. 

214. The sum of any even powers of two numbers, as a^+V, 
a* + &*, a® + 6^ (a^" + b^), may be factored by adding to the 
binomial such a number as will make it a perfect square, and 
then subtracting the number added from the result, as shown 
in Case V., §§ 194, 195. 

Thus, a* + &* = a* + 6* + 2 a'b^ - 2 aV 

= (a2 -{-by -2 aV = (a" + b^ + V2a^^ (a* + ft^-. VSo^p). 

Resolve into factors 

23. a;2 + 2/^. 27. a^ + b\ 

24. a^ + ft*. 28. 9a;* + 42^. 

25. m*4-w®. 29. 4ar^ + y*. 

26. 81 + aJ*. 30. a^4-256. 



§ 216.] FACTORING. 113 

FACTORING BY SYNTHETIC DIVISION. 

215. It will be found by multiplying together several poly- 
nomial factors, arranged according to a common letter or 
letters, that the first term of the product is the product of the 
first terms of the factorSy and that the last term of the product 
is the product of the last terms of the factors. 

Thus, (aj-2) (a; + 3) (aj-5) = aj8-4aj2- llaj-f 30; and 
(a? 4- a) (x — b) (x + c) =a^ + (a — b + c)a^ — (ab — ac -|- be) x 
— abc, 

216. It follows that a polynomial which contains one or 
more binomial factors may be factored by resolving its first 
and last terms into factors, and then determining by synthetic 
division (§ 132) which of the binomials that can be formed by 
uniting these factors (two and two) with the sign + or — 
are factors of the given polynomial. 

This process is made plain below by the factoring of a 
few polynomials, and a little practice will enable pupils to 
use it readily. 

1. Resolve into factors aj" — 4 aj* — 11 a? 4- 30. 

The three factors of a^ are x, x, and a?, and the three positive 
factors of 30 are 2, 3, and 5, or 1, 5, and 6, or 1, 2, and 16. 
Since two of the terms of the polynomial are negative, and 
the last term positive, try successively by synthetic division the 
divisors a; — 2, a? + 3, and x — 5. 



1 

+ 2 


1_4_114.30 

+ 2 




- 4 




-30 


1 
-3 


1-2-16 
-3 




+ 16 




1-6 



(x2 _ 2a5 — 15, first quotient) 



(x — 5, second quotient) 

Hence the factors sought are x — 2, x + 3, and x — 6. 
white's alo. — 8l 



114 



ALGEBRA. 



[§ 216. 



If we had tried as divisors aj-flora? — 1 or a;+-2ora? — 3 
or x + 5y there would have been in each case a remainder, thus 
showing that the trial divisor is not a factor. 

In the foregoing process it was assumed that the polynomial x* — 4x> 
— 11 X + 30 is composed of three binoinial factors ; but it may be found 
on trial that a polynomial of the third degree contains only one rational 
binomial factor, the other factor being a trinomial, as in Example 5. 

2. Eesolve into factors a:? — 43 a; ■+ 42. 



Try X — 1, a — 6, and x + 7. 



+ 1 



1 
+6 



l-_0-43+42 

+ 1 
+ 1 
-42 



1 + 1-42 (x2+x-42,lstquo.) 
+6 

+42 



The process at the left may be 
condensed by writing the several 
partial products in a horizontal line, 
thus: 

l_0-43+42 
+1+ 1-42 



+ 1 
1 

+6 



1+1-42 (x2+x-42,lstquo.> 

+6+42 
1 + 7 (x + 7, 2d quo.) 



1 + 7 (x+7, 2dquo.) 
The factors sought are x — 1, x — 6, and x + 7. 

3. Resolve into factors a?* - 10 ic* + 32 ar^ - 38 a? + 15. 

First try in succession x — l,x — 3, x — 5. 

1-10 + 32-38 + 15 
+ 1 - 9 + 23-15 

(x8 - 9 x2 + 23x - 15; 1st quo.) 



1 

+ 1 
1 

+ 3 

1 

+ 6 



1_ 9 + 23-15 
_l- 3-18 + 16 



1- 6 + 
+ 6- 



5 
5 



(x2-6x + 5, 2dquo.) 



1—1 (x — 1, 3d quo.) 
The factors sought are x — 1, x — 3, x — 5, and x — 1. 

4. Resolve into factors 2 ar* + 31 a^ + 62 a? - 39. 
Try in succession 2 x - 1, x + 3, x + 13. 



2 

+ 1 

1 

-3 



2 + 31 + 62-39 
+ 1 + 16 + 39 


1 + 16 + 39 
- 3-39 




1 + 13 





(x2 + 16x + 39) 



(X + 13) 
The factors are 2 x — 1, x + 3, and x + 18. 



216.] 



FACTORING. 



116 



5 . Resolve into factors 3 ic* — 46 05* + 9 a? + 2. 
First try 3 x-l. 



3 

+ 1 


3-46+ 9 + 2 
+ 1-16-2 




1-15- 2 



(a;2-15x-2) 

If we try aj — 1 or a? + 1 or a? — 2 or a; -f 2, we shall have in 
each case a remainder, and so conclude that aj* — 15 a? — 2 can- 
not be resolved into rational factors. This is also made obvious 
by the fact that the coefficient of the second term (— 15) is not 
the algebraic sum of any two rational factors of — 2 (§ 371). 

Hence the factors sought are 3 a? — 1 and a?* — 15 aj — 2. 

6. Resolve into factors aj*- 3 a?*- 18 aj8+ 87 ^— 109 x + 30. 

Try first successively x — 2, x — 3, and x + 5. 

1 



+ 2 

1 
+ 3 

1 
-6 



1-3-18 + 87-109 + 30 
+ 2- 2-40+ 94-30 



l_l_20 + 47- 16 
+ 3+ 6-42+ 15 



1 + 2-14+ 6 
-6+16- 6 



I 1-3+1 



(x2_3«+l) 



It will be found by trial that neither a; — 1 nor » + 1 is a 
factor of a^ — 3 a? -h 1 ; and it is also evident on inspection that 
the trinomial has no rational factors. 



Resolve into factors 

7. aj'-f-4ar*-f aj-6. . 

8. aj8-7aj-6. 

9. aj»-19a?-30. 

10. aj8-43aj + 42. 

11. a^'-lOaj^-f 31aj-30. 

12. a^ + 9a^-f-ll«-21. 

13. aj8-a2-8aj + 12. 



14. 3a:»-10aj*-aj + 12. 

15. 2aj8-9a^ + 13aj-6. 

16. 6a^+7aj+-13. 

17. 2a^-5aj + 39. 

18. aJ*-4a^-7aj« + 34aj-24. 

19. aj* + a^-4a?-16. 

20. aJ*~3aj»-3ar^ + 12aj-4. 



I 



118 ALGEBRA. [§ 217. 

67. a^ — 64. 83. a^ — 27 x -\- ISO, 

68. 1 — m«. 84. ic' — aj2 — 20a?. 

69. 27-»'. 85. 4a*-14a + 6. 

70. l-\-(x + yy. 86. 4a* — 10a* 4-4. 

71. (x-yy-\-l. 87. 3aj*-2a;-65. 

72. aJ* - 12 aj2 + 11. 88. 3a;2-6a;-15. 

73. a?* — 8 a;* + 7. 89. a?* — 6 a? — 7. 

74. a*-10a2 + 9. 90. a:*-10aj + 25. 

75. a;*4-2a;-120. 91. »«-5aj-14. 

76. a:2 + 21aj-72. 92. a?*-7a;2-60. 

77. aj2 + a?-240. 93. a;* -6ar^- 40. 

78. a:* _ 10 aj _ 144. 94. 81 a?* - 72 »y + 16 j^. 

79. a;2 + 30 aj + 221. 95. a^ + m^a^ - n^a^ - nM. 

80. 21ar^-4a;-l. 96. a;2^_2aaj + 2aj+(a4- 1)'. 

81. 36a;2-31aj-56. 97. a* - a^y + af - y^. 

82. ar^ - 41 a; + 420. 98. aJ* -(3m - 2)a;* - 6m. 

99. ar^ + 2/* + 2*-f2ajy-2a»-2y2;. 

100. a" -{- b^ - (^ - <P - 2 ab -2 cd. 

101. m^ + n^ - a^ - b^ - 2mn + 2al). 

102. 4 — 4a + 2mii — m^-l-a*- nl 

103. 9a^-{-f-4:Z^-6xy-Ssz-4:S^. 

104. a^4-2/^ + l-2a;y + 2aj-22^. 

105. aa^ — by^ + 2 aa^y — 6aj* -f- aosy^ — 2 bxy, 

106. aV-2a6a^-|-&^a^-aY + 2a62/*-&V. 

107. ma^ + 2 na^ — nyf — 2 mxy — no? + wiy*. 

108. aj3-9a^ + 23a;-15. 

109. o^-2a^-^iea^-\-2x + 15. 

110. a^ + 7aj' + 9a:2 .7aj_10. 



§221.] COMMON FACTORS. 119 



CHAPTER VIII. 

COMMON FACTORS AND MULTIPLES. 

COMMON FACTORS. 

218. Any number is divisible by any one of its factors or 
by the product of any two or more of its prime factors. Thus, 
since 30 = 2 x 3 x 5, 30 is divisible by 2 or 3 or 5, or by 2 x 3 
or 2 X 5 or 3 X 5. In like manner 3 ab is divisible by 3 or a 
or 6 or by 3 a or 3 6 or ab, 

219. Two or more numbers are divisible by any common 
factor or by the product of any two or more common prime 
factors. Thus, since 12^= 2 x 2 x 3, and 30 = 2 x 3 x 5, 12 
and 30 are divisible by 2 or 3 or by 2 x 3. In like manner 
3 a^b and 6 abc? are divisible by 3 or a or 6, or by 3 a or 36 
or ab or 3 ab. 

220. It follows that a common factor of two or more num- 
bers is their common divisor. 

Two or more numbers which have no common integral fac- 
tor except 1 are prime with respect to each other. 

Highest Common Factor. 

221. The highest common factor of two or more numbers is 
the factor of highest degree that will divide each number with- 
out a remainder. Thus, ± 3 a^b is the highest common factor 
of 6 a^bx and 9 a^b^y, and ± 4 a^b^ is the highest common factor 
of 12 a^ft^ and - 16 a^b\ 

In arithmetic the greatest number that will exactly divide two or more 
numbers is called their greatest common divisor ; but since, in algebra, a 



120 ALGEBRA. [§ 222. 

negative number is less than any positive number or than a negative 
number that is numerically smaller (§ 67), the highest common factor 
of two or more numbers may be algebraically less than any other fac- 
tor, positive or negative. Thus, — 3 a*6 < ± a or ± ah. This explains 
and justifies the use of the expression ** greatest common factor '' in arith- 
metic, and ^* highest common factor '^ in algebra. 

222. The highest common factor of two or more algebraic 
numbers is their highest common divisor. 

For convenience the abbreviation H. C. F. is used in this 
chapter for " highest common factor," and H. C. D. for " high- 
est common divisor." 

H. C. F. FOUND BY Factoring. 

223. The H. C. F. or H. C. D. of two or more algebraic 
numbers may be found by resolving each into its prime factorSy 
and then taking the product of all those factors which are common. 

The numbers in the following exercises may be readily 
resolved into factors by inspection, 

1. Find the H. C. F. of 63 a^a?y and ^abn^. 

63a2a;83/ = 3 x3x la^o^\ 
42 ahx^z^ = 2x3x7 abx^z^, 
. % 3x7 ax^, or 21 ax% is the H. C. F. required. 

The H. C. F. of two or more numbers may be either positive or neg- 
ative, and hence ± 21 ax^ is the H. C.F. of eSa^hfiy and 42abx^z^, In 
this chapter only the positive H. C. F. is given or required. 

2. Find the H. C. F. of ar^ - 0^2^ and aS - 2 iB«3/ -h ojy*. 

ofi-xy^ = x (pfi -y^)z=ix(X'^ y)(x - y); 
aj8 _ 2 x^ -^xy^ = x(x- y) (x - y), 

.'. x{x — y), or x^ _ xy, is the H. C. F. required. 

3. Find the H. C. F. of a^-12x-\-S5, aj^-10a? + 26, and 

aj* + 6 aj — 55. 

x2 _ I2x + 35 =(x - 5)(x - 7) ; 

x2 _ lOx + 25 =(x - 5)(x - 6) ; 
x^^ 6x-56=(x-5)(x+ll). 
.*. X — 5 is the H. C. F. required. 



§223.] COMMON FACTORS. 121 

4. Find the H. C. F. of a* - 2 a^h^ -+■ 6* and a* - h\ 
a* - 2a262 + 6* = (a2 - h^y = ia^ - 62)(a2 - fts) . 

.-. a2 - 62 is the H. C. F. required. 

6. Find the H. C. F. of 3a^ + 4aj - 15 and 5aj* - 56aj + 11. 

3x2+ 4x-16=(3x-5)(x + 3)j 
6x2 - 56x + 11 =(6x - l)(x - 11). 

There is no common factor, and hence no H. C. F. 

Find the H. C. F. of 

6. 210 and 330; 450, 625, and 825. • 

7. 21a86^ and 63a6»c; l^d'ly'c and 60a6V. 

8. 2^0? f 7?, IGajy, and 12w^t^. 

9. ^a^a?y^, 24:a^ahf^y 72 a»a^/, and 36 aV^^. 

10. aj" — ^, a^ — ]^, and a^ — 2xy + i^. 

11. aj" - 1, (x- 1)2, and a^-1. 

12. a^ 4- aV and aJ* — a^a^. 

13. 15 (a? - ly and 45 (a?* - 1). 

14. 4 (a« - b^ and 20 (a* - b% 

15. a^ + &* and a^ - a^b -f ab\ 

16. a* — 2aj + l and a^ — 1. 

17. a^ + aj + 1 and a^^ + aj^ + l. 

18. a^-8a^ + 16 and 3a^- 12a^ + 12aj. 

19. 10 ay + 40 aV + 40 aV and y* - 16«*. 

20. aV — ay and ar^ + a^y, 

21. aj* 4- 8 a? -I- 16 and a?* - 256. 

22. a^ - 10a; + 9, ar* - 17a; + 72, and a;^ - 11 a; + 18. 

23. a;* + 10aj-|-21, a^-\-4:X — 21, and ar* — a; — 56. 

24. «» - 9a; - 36, 2a;* - 30a; + 72, and 3a;2 - 42a; + 72. 

25. 3a^-4a;-7, 5a;»H-3a;-2, and 15 a;* + 18 a; + 3. 

26. a;* + 3;* -132 and a;* + 17 a;* + 60. 



122 ALGEBRA. [§ 224. 

27. a^-llaj»-f-10a^ and ic* + 7aj — 8. 

28. «* — 18aj + 65 and a* + 2aj — 35. 

29. 6a? + x — 12 and 4:a^ + 12x + d. 

30. a? — ly iC* — 1, and q? — 2qi? + x. 

H. C. E. FOUND BY Continued Division. 

224. When two or more polynomials are not readily resolved 
into factors by inspection, their H. C. F. may be found by the 
method of continued division^ a method similar to the corre- 
sponding one in arithmetic. 

225. This method depends on simple principles : to wit, 

I. The factor of a number is the factor of any multiple of the 
number. Thus, 6, a factor of 12, is a factor of 12 x 2, 12 x 3 ••• 
12 X n. 

II. A common factor of two numbers is a factor of their sum 
or of their difference, also of the sum or difference of any of their 
multiples. Thus, 6, a common factor of 24 and of 18, is a 
factor of 24 + 18 or 24-18, also of 24x2 + 18x2 or 24 x 
2 - 18 X 2. Hence 

III. A common factor of either of two numbers and their 
difference is a common factor of the two numbers. Thus, 6, a 
common factor of 18 and of 30 — 18 (or 12), is a common 
factor of 18 and 30. 

1. Find the H. C. F. of 288 and 816. 

288)816(2 
676 

240)288(1 
240 
48)240(6 
240 

48 is the H. C. F. of 288 and 816. 



§226.] COMMON FACTORS. 123 

Since 48 is a common factor of 48 and 240, it is a factor of 
288, their sum (II.). 

Since 48 is a factor of 288, it is a factor of 288 x 2, or 576 

(I.). 

Since 48 is a common factor of 240 and 576, it is a factor 
of 816, their sum (II.). 

Hence 48 is a common factor oi 288 and 816. 

Since any common factor of 288 and 816 is a factor of 240 
(II.), and hence a common factor of 288 and 240 and also 
of 240 and 48, it follows that 48, the greatest common factor 
of 240 and 48, is the H. C. F. of 288 and 816. 

226. The following is a general explanation of this method : 
Let a and h represent the two numbers, and g, q\ g", and 
r, r', r", and so on, denote respectively the successive quotients 
and remainders, and suppose that r" = 0. Thus, 

h)a{q 

r)b(q^ 
rq^ 

r')r(q'^ 
rY 

Since r' is a common factor of r' and r, it is a factor of rq' 
(§ 225, 1.), and hence of 6, the sum of r' and rq' (II.). 

Since r' is a common factor of r and 6, it is a factor of bq 
(I.), and hence of a, the sum of bq and r (II.). 

Hence r' is a common factor of a and b. 

Since any common factor of a and & is a factor of r, and 
since any common factor of b and r is a factor of r\ it follows 
that the greatest common factor of r and r' (which is r') is the 
H. C. F. of a and b. 



124 



ALGEBRA. 



[§ 227. 



2. Find the H. C. F. of 6a?-7a^^22x+S2 and Saj^+aj-lO. 



6a;8_7a;2_22x + 32 
6«8 + 2aca-20a; 



3x2 + 35-10 



2«-3 



-9x2-2x + 32 
-9x2-3x-30 


3x2 4-x-10 


X4-2 


Sxa + ex 


3x-6 


-6X-10 
-6X-10 





.*. X + 2 is the H. C. F. required. 



3. Find the H. C. F. of a^-a^b-{-ab^-b* and a^-a^h-ab^-^-l^. 



a* - a^b + ab» - 6* 
a* - a8& - a262 + ab^ 



a9 - a% - a&2 + 58 



a 



a^b^ - b* = (a^ - b^)b^ 



€fi 


- a^b - ab^ + b^ 
-ab^ 


a^-b^ 


cfi 


a-b 



-a^b 



+ 68 
+ 68 



a2 - 62 is the H. C. F. required. 



227. It is seen that a^b^ — b\ the first remainder, will not 
divide the first divisor; but aV — b^ = (a^ — b^b^, and, remov- 
ing the b% we obtain a^ — 6^, which is a divisor. Since b^ is 
not a factor of a^ — a^b — ab^ + b% it is evident that it is not 
a factor of the H. C. F., and hence may be set aside. At any 
stage of the process, a factor of dividend or divisor, not com- 
mon to both, may be rejected. 

4. Find the H. C.F. of 3a^+a^-4aj-10 and 9a^-9x-10. 
Multiply by 3, 3x8+ x2- 4x-10 



9x8+ 3x2- 


- 12 X - 30 
-lOx 


9a;2_9a;_io 


9x8- 9x2- 


x + 4 


Multiply by 3, 12 x2 - 


- 2X-30 


36x2- 


- 6X-90 


(9x2-9x-10)x4 = 36x2- 


-36X-40 


Divide by 10, 


30X-50 


9a;2_- 9x-10 


3x-6 




9x2- 16 X 


3x + 2 


6X-10 




6X-10 


.-. 3a 


; - 6 is the H. ( 



§230.] COMMON FACTORS. 125 

6. Find the H. C.F. of 10 aho^ -^ 10 abx^ - 90 a^x - 90 ab 
aaid 6 aod^ — 42 aot^ — 36 ax, 

10ab3fi + lOabx^ - 90 a6x - 90 a5 = 10ab(iifi -^x^-dx- 9); 
6 aaj* - 42 ax^ - 36 ax = 6ax(sfi - 7 x - 6); 
10 ab = 2 a X 6b ; 6ax = 2ax3x. .*. 2a is common. 



x8 + x2-9x-9 
x8 -7x-6 



a*_7x -6 
x8_2x2-3x 



x8-7x-6 



x^ — 2 X — 3, common. 



X +2 



2x2-4x-6 
2x2-4x-6 

.-. 2a(x2 - 2x- 3) is the H. C.F. required. 

228. At any stage of the process, a divisor or dividend may 
be multiplied or divided by any factor that is not common to 
both, as in Example 4; and any common factor may be set 
apart as a factor of the H. C. F., as 2 a in Example 5. 

229. To find the H. C. F. of two polynomials by continued 
division, 

Remove the monomial factor, if any, from ea>ch polynomial^ 
and set aside the common factor in the same, if any, as a 
factor of the H. C, R 

Arrange the resulting polynomials in descending powers of 
some common letter, and divide the polynomial of the higher 
degree by the other, and then the divisor by the remainder (if 
any), and the second divisor by the second remainder, and so on 
until there is no remainder. The last divisor is the Ei, O. F. of 
the first dividend and divisor, and the product of this factor and 
the common monomial factor, if any, is the H. C. F. required. 

If the two polynomials are of the same degree of the common 
letter, either may be used as the divisor. 

230. To find the H. C. F. of more ^han two polynomials, 
First find the H, C. F. of two of them, and then the JET. C, F. 

of this result and a third polynomial, and so on. 



126 ALGEBRA. [§ 231. 

Find the H. C. F. of 

6. a* - 18a; + 65 and a^ - 18aj - 35. 

7. 6«2 + a;-12 and 6a»H-7a*-aj + 3. 

8. a:« + 125 and 2a8 + 7»2 + 75. 

9. ar* -lOaj -24 and »«_ 23a; 4- 28. 

10. ar' -16aj-t- 21 and a;3_j_ 7^ _ 43 

11. a:S + 15aj-306 and a^-26aj-60. 

12. 4:a^-21a^^l5x-\-20 Biiid a^^6x-i-S. 

13. 2a;«-120aj-h378 and 5a;«-42a;2^73^_,_g3^ 

14. aj* — a:* + 2a;2 -ha;-f 3 and a^ + 2ar* — a;-2. 

15. ar* + 3a* + 4a; + 12 and a;» + 4a;2 + 4aj + 3. 

16. 4aJ*-8a:3-20a;2_,_24a? + 20and3a» + 6a*-24aj-45. 

17. a^-5ix^-\-Sa^-7x-\-3 Qnd 2a^-9a^-^10x-3. 

18. 18a«-51a;* + 13aj-f 5 and 6a;2_i3a._5 

19. 2a;*-3a,'«4-2a;2_2a;-3 and 6ar*-«2 + 8aj + 3. 

20. a^ — a^y — Qcy^ + y^ and a^ -^ x^y — xy^ — t^. 

The H. C. F. of the polynomials in the foregoing examples may be 
readily found by synthetic division (§ 216). It is recommended that 
the polynomials in several of the foregoing examples be thus resolved 
into factors, and their H. C. F. found. 

COMMON MULTIPLES. 

Lowest Common Multiple. 

231. A multiple of a number is the product of the number 
multiplied by an integer ; and hence any multiple of a num- 
ber is exactly divisible by the number. 12 is a multiple of 2, 
3, 4, and 6 ; and 3 a& is a multiple of 3, a, and b, also of 3 a, 
3 bf and ab. 

The product of two or more integral factors is a multiple of each factor. 



§ 235.] COMMON MULTIPLES. 127 

232. A common multiple of two or more numbers is a mul- 
tiple of each of them. Thus, 12 is a common multiple of 2, 3, 4, 
and 6 ; and 6 a%^ is a common multiple of 6, a', b\ 3 a*, 3 b^, 
etc. ; and aj* — ^ is a common multiple oi x + y and x — y, 

233. The lowest common multiple of two or more numbers is 
the lowest multiple of each of them. Thus, 12 is the lowest 
common multiple of 3 and 4, and a^ — ^ is the lowest common 
multiple oi x + y and x — y. Hence 

234. A common multiple of two or more numbers is exactly 
divisible by each of them; and the lowest common multiple of 
two or more numbers is the lowest number that is exactly divisible 
by ea>ch of them. 

For convenience the abbreviation L. C. M. is used for 
** lowest common multiple." 

The L. C. M. found by Factoring. 

235. Since any multiple of a number contains all of its 
factors, a common multiple of two or more numbers contains 
all the prime factors of each ; and the L. C. M. of two or more 
numbers contains oil the prime fojctors of each number, and each 
factor in the highest degree in which it occurs. 

Thus, the L. C. M. of 3 a^b^ and a%V is 3 x a^ x 6* X c*. 

1. Find the L. C. M. of 6 a^ba^ and 10 ab^a^f. 

10 a62a;3y2 = 2x6 ab^y^. 
.'. 2 X 3 X ^a%Vy\ or ^Oa'^b'h^y^, is the L.C.M. required. 

2. Find the L. C. M. of a?-f, ^-2<x^-\-f, and 3X^+2 icy+2/l 

a;2 _ y2 _ (aj + y) (x - y); 
x2-2a;2/ + y2=(a5--y)2; 
»2 + 2jcy + y2 = (a; + y)2. 
.'. (« + yY X (ac — y)2, or a* — 2 x2y2 ^ y4^ jg the L. C. M. required. 



128 ALGEBRA. [§ 236. 

8. Find the L.C.M. of a^-9, 15 a^ - 39 a? - 18, and Saa^ 
-18 oa? 4- 27 a. 

x^ - 9 =(x + S)(x - S); 
15x2 - 39x - 18 = 3(x - 3)(5x + 2); 
3a2x2-18aa; + 27a=3a(x-3)2. 
.-. 3 a(x - 3)a(x + 3) (6 x + 2) is the L. C. M. required. 

Find by factoring the L. C. M. of 

4. 5a^x and 15 a^ocih/, 

5. Uafyn^ and 63an^s^, 

6. 6a^c(P, Oa^ftV, 15a6V, and 20aVdaj». 

7. 01^ — y^ and a^ — i^. 

8. 7x(a+by and 5y^(a^-^lf). 

9. a?-'a% a^ — b% and a* — 5a6 + 4&'. 

10. ar*-7aj + 10 and a:2_|_33.__ j^Q 

11. a^^-aj-llO, aj2 + aj-90, and 2a? -{-15x — BO. 

12. aj» + 8aj + 15 and aj2 + 4a; — 5. 

13. aj2 - 5aj - 84 and a^-7aj- 60. 

14. 7aj2_36a; + 5 and a^-25. 

15. a^ + od, ab — b^, and a^ — 6*. 

16. aj* — y*, a^ — y^, a^ + 7/^, and a?* — 2ajy + 2^. 

17. a* - 1, a^ + 1, ar^ - 1, and a^ - 1. 

The L. C. M. found by the H. C. F. 

236. When the factors of two polynomials cannot be found 
by inspection, they may each be resolved into two factors by 
Jiiiding their H, C. F. by continued division, and then dividing 
ecich by this H, C. F. 



§238.] COMMON MULTIPLES. 129 

237. The H. C. F. of two polynomials contains cUl their com- 
mon factors, and the two quotients obtained by dividing each 
polynomial by the H. C. F. contain all the fa/stors not common ; 
and hence the product of their H. (7. F, and the two quotients thus 
obtained will he the L. C. M, of the two polynomials. It contains 
all of their factors, and each in the highest degree. 

The process may be somewhat abridged by dividing one of the 
polynomials by their H. O. F,, and multiplying the quotient by the 
other polynomial, 

1. Find the L. C. M. of a^ — a^ ^ xy' + 1^ and a? + a?y 
— xy^ — f. 

The H. C. F. of the polynomials is x^ - j^, 
and (pfi — xhf — xy^ + y^^-^x^ — tfl^x — y, 

,; (x — y) (a^ + x^ — xy^ — y^)m the L. C. M. required. 

2. FindtheL.C.M. of or* -48 a? 4- 7 and a^ + 8aj* + 9aj + 14. 

The H. C. F. of the polynomials is x + 7, 
and («8-48iB + 7)-T-(x + 7) = x2_7a;4.i. 

.-. («2 - 7x 4- l)(x' + 8x2 + oaj + 14) is the L. C. M. required. 

Find the H. C. F. and the L. C. M. of 

3. 12a^-4aj-21 and 6a^-17aj + 12. 

4. aj*-23a^-10a? and 7a^-34ar^-4a?-5. 

5. 3ar»-8a^-12a;-l and 5aj8-12aj2-24a;-7. 

6. So? -5x + 2 Bji^ 21a^-20a^-llx-{- 10. 

7. 4aj»-3ar*4-16a:-12 and 12aj8- 17 a^ + 22 aj- 12. 

8. ar^-2a^-a^ and aj» + 2i»2 + 2a; + l. 

9. a3-9aj2-f26a;-24 and a^-6a^ + llaj-6. 

10. 6aj2-13a;+6, 2ar* + 5aj-12, and 6i»2-a;-12. 

238. The L. C. M. of the polynomials in the above examples 

may be readily found by synthetic division, the factors thus 

obtained being united as in § 235. 
white's jllo. — 9 



130 ALGEBRA. [§ 239. 



CHAPTER IX. 

FRACTIONS. 

239. A fraction is one or more of the equal parts of a unit. 
The unit which is divided into equal parts is the unit of 
the fraction, and one of these equal parts is the fractional unit. 

An integer is one or more integral units, and a fraction one or more 
fractional units. 

S840. A fraction is expressed by two numbers, — one called 
the denominator, denoting the number of equal parts into which 
the unit is divided ; and the other called the numerator, denot- 
ing the number of equal parts taken. 

241. In arithmetic, fractions are expressed by words or 
figures. When expressed by figures in the common form, the 
numerator is written above, and the denominator below, a 
horizontal line, as ^. In expressing decimal fractions, the 
denominator need not be written, but may be indicated by the 
decimal point. Thus, seven tenths is written ^ or .7. 

In algebra, fractions are expressed by algebraic symbols, the 
numerator being written above the denominator, as in arith- 
metic. Thus, if a unit is supposed to be divided into 6 equal 
parts, and a of these parts are taken, the resulting fraction is 

expressed by -. 

In both arithmetic and algebra the numerator and denominator may be 
separated by an oblique line (called the solidua)^ as ^/^ and " A (§ 20). 



§ 249.] FRACTIONS. 131 

242. In algebra a fraction is treated as an indicated division^ 
the numerator being the dividend^ and the denominator the 
divisor. Hence 

243. An algebraic fraction is the quotient indicated by writing 
the dividend over the divisor, with a line between them. 

Thus, the fraction - is the quotient of a divided by b, and 

is read " a divided by 6." 

244. The numerator and denominator are called the terms 

of a fraction. Thus, a and a -f 6 are the terms of — ^. 

a-to 

245. Since a-5-&=ac-f-6c, - = — ; and since ac-i-bc=a-^b, 
^« ^ b be 

^ = 2. Hence 
be b 

Both terms of a fraction may be multiplied or divided by the 
same number without altering the value of the fraction. Hence 

246. The same factor in both terms of a fraction may be can- 
cded, or the same factor may be inserted in both terms, without 
altering the value of the fraction, 

247. Sinceg = ^^(-^) =^and:ig= ^^^(^^) =g, 

b bx(-l) -b' -6 _ftx(-l) b' 

it follows that the signs of both terms of a fraction may be 
changed without altering the valu£ of the fraction, 

248. 2 or ^^= +% and ±5 or ^^= -2; and hence a 
b — b b —b +6 

fro/ction is positive when the signs of both its terms are alike, and 

negative when the signs of both its terms are unlike. Hence 

249. The changing of the sign of either term of a fraction 
changes the sign of the fraction. 



132 ALGEBRA. [§ 250. 

Multiply both terms of 

A. H (jL a X OS 

1. -, — , -, -, -, and by 3. 

3 5 m^ m 3 

a + b 5 m + 1 a; + y 1+y 

Divide both terms of 

. 6 9 3a 6 3a „, 3,0 

*• IK' To' ~^' To~' ^T' ^^^ ?r~i "y ^' 
15 12 6 12a 6 6 9nr 

6- ^ 2 > -^ . o , o > and ^ . bya?+«. 

a^ — y^a^-\'2ocy-\-y^ x-\-y -^ ^ 

a2-&2 a3-_&3 a*- 6* , , 

6. , , and bv a — o, 

3a2-3a6' a'-b^ a'-2ab + b' y ^ ^' 



REDUCTION OF FRACTIONS. 
Case I. 

250. Fractions reduced to lowest terms. 

A fraction is in its lowest terms when its terms contain no 
common factor, i.e., are prime to each other (§ 220). 

3 a^b^G 
1. Reduce — to its lowest terms. 

15a6V 

3a2&8c Sab^Gxah ab 



16a62c8 8a62cx5c2 6(J2 

The common factors of the two terms are 3, a, &^, and c, and are 
canceled. 

a* — 6* 
2. Reduce — r to its lowest terms. 

a* + 2 d'b^ -h 6* 

a4 _ 54 ^ (gi 4. 52)(q2 _ 52) ^ q2 _ yi 
a*4-2a262 + 64 (a2 -|- 52) (a^ + &2) a^ + ^' 

The common factor is a2 4- b^, which is canceled. 



S 250.] FRACTIONS. 188 

3. Reduce ^-1^^ + ^^ to its lowest terms. 

a^ — 12 0? 4- 35 

a;« - lOx + 26 _ (g - b)(x - 6) _ a5 - 6 
aj2 - 12 X + 35 (x - 6)(x - 7) x - 7* 

4. Eeduce ^^ - ^^^ ^^I'^K to its lowest terms. 

a* - a^ft + ab^ - b^ 

(cfi-a2b^ab^ + J)^)-i-(a^-b^) ^ a-b 
(a* - a^b + ab* - 6*)-^(aa - 6-2) a^-ab + b^ 

The H. C. F. of both terms, found by inspection or by continued divis- 
ion, is a^ — b^; and dividing both terms by a* — 6^ reduce* the fraction 
to its lowest terms. 

Reduce to lowest terms 

6. l^^^. 16. ^ + 2^ 



25 ab*c a?* + icy + 2^ 

^ lOS a^b^x ,^ aj8-1728 



144 a& V ^2 _ ^ _ 132 

- — 91 m*p^aj* -^ ar*+(a-|- 6)a;4-a5 
119 mpY^ ' ' aj2+(6-hc)a;-|-6c' 

g a* — a?h 7? — {a — b)x—db 

Sab ' a^ -\-(b ^c)x — bG 

^ 12 afxy ^^ ^-t ^ 

^aa? — 2Aay ' oc^ — y^ 

10. ^^(^-^)' 20. ^'-"< 

„ aj2 4-13a; + 22 ^_ a^-aaj^ 

11. — t: _ • m1. 



aj2 + 2a?-99 a^-2aa;+a2 

12 .^ + ^^-^ . 22. «' + ^ 



flj8 _ 14iB2 4- 49a; a^ + 2a^x + aa? 

^3 g» 4- a; - 72 ^^ 3 x^ ^ 35 ^jj _^ 105 a; 



4aa^-48aa;4-128a 12 a;^ ^ 132 a;2 _^ 3^0 a; 

a«2-15a? + 54 2 a^ + 19 a; + 35 

' a»-18ar*4-101a;-180' * 3 a;^ 4- 15 a? - 42* 



184 ALGEBRA. [§ 251. 

a^-^ 47 a? 4- 14 g^ a? + 125 

' a^~54aj-35* ' 23^ + 70^ + 76 

2a.-«-13a^ + 23a;-12 o^- 16a; + 21 

* 7a^-33a2^18a; + 8* ' a^ + 7aj-48' 

a^-18a; + 35 a^ + 30;^ + 4fl? + 12 

 a^-55x + 126 ' ix^ ^4.0^ -\- 4:X + 3 

a*-a2a; + 3a^-3ar' ^^ a^-a^ + 2o^ + x-]-S 

2o. • o«5* — » 

4 a^a? — aic^ — 3ic' a^ + 2a:^ — x — 2 



29. 



a^- 135a; -486 ^^ a^ - a^ - xf -\- f 

a;* — 24 a;* — 63 a; — 54 * a;* + a;^ — a«^ — 2/* 



261. To reduce a fraction to its lowest terms, 
Resolve both terms of the fraction into their prime f actor Sj and 
cancel all the common factors; or divide both terms of the frac- 
tion by their JET. C F, 

Case IL 

252. Fractions reduced to integral or mixed numbers. 

An algebraic mixed number is an integer and a fraction con- 
nected by the sign + or — . Thus, «^ + j and a^ ^— - are 

mixed numbers. ~" 

In arithmetic a mixed number is an integer and an added fraction. 
Thus, Q\-Q^\, and 16f = 16 + 1. In algebra the integral and frac- 
tional parts of a mixed number may be connected by + or — . 

253. Since a fraction is an indicated division (§ 243), it is 
reduced to an integer or a mixed number by performing the 
operation indicated. 

In arithmetic every improper fraction can be reduced to an integer or 
to a mixed number, but in algebra this can be done only when at least 
one term of the numerator is divisible by a term of the denominator. 

Thus, --=- cannot be reduced ; but 
b 

= 2 a — + , anA = a + 0. 



2a + 6 2a+6 a-b 



255.] FRACTIONS. 186 

1. Reduce — — — -^ — to a mixed number. 

= X — a + -. 

x^ X 

2. Reduce -^^ to a mixed number. 

JC— 1 05 — 1 

Reduce to integral or mixed numbers 

^ aa? — bx ^^ a^ — Sx — B 

3. . 10. 5 . 

X x — l 

^ a^ + x ,, a^ — a^ + 3c 

4. • 11. . 

a a + x 

abc — CO? -^o^ + Sajy — 5 

5. • 12. : • 

ac aj + y 

^ a?*-3^ -„ 3a^-15aa^4-a' 

6. — :; ;;• Xt5. ' ^ • 

a^ + .y* 3a? . 

a; + 3^ * 4:0^ — aP — 4:X + 1 

a^-y« a^-3a^6 4-3a6^-y 

®' aj-2^" • a^^2ab + b^ 

9. — -^- 16. 

a — b ic — y 

254. To reduce an algebraic fraction to an integral or mixed 
number, 

Divide the numerator by the denominator. 

Case III. 

255. Integral or mixed numbers reduced to fractions. 
1. Reduce 16^ to a fraction. 

i«j 16 X 3 ,2 48 + 2 60 
^^ = -3— ^3=-3- = T 



186 ALG£BBA. [§ 256. 

2. Reduce a — r to a fraction 

o-f 6 

a« a(a + b)-a^ db 

d := — ^ ^ • 

a + b a -\- b a + 6 

3. Reduce x-^y toa, fraction with x — y for its denominator. 

-^ . „ _ (a^+y)(x-y) _ x^-^ 
x—y x — y 

4. Reduce a — h r-to a fraction. 

gg + ft' ^ (g - b){a + 6)-(a« + &') ^ a' - 52 - ga _ &a _ -26«' 
" g + 6"" g + 6 ~ a + 6 a+b 

Reduce to a fraction 

6. 3i»H--. 10. a^ — h^ -^. 

6. 6ay + i^^^^. 11. aj«-ajy + 3^- ^^ 



y » + y 

7. l_^. 12. {x^zy+ ^ 



x-\'y ^ ' x — z 

Sax — a ,«o o Sa^ — 2^ 

8. 5a ^— 13. Sx — 2y  — ^. 

x — 1 i» + y 

9. a4-» V — 14. ; ab + 2. 

a-\-x a-{- X 

266. To reduce an integer to a fraction, 

Multiply the integer by the denominator of the required frao 
tion, and under the product wrUe the denominator. 

267. To reduce a mixed number to a fraction, 

Multiply the integral part by the denominator of the fraction, 
and to the prodv/st add, or from the product subtract, according 
as the fraction is + or —, the numerator, and under the resvU 
ivrite the denominator. 

When the fraction is preceded by the sign — , as in Example 4 above, 
either its numerator, if a polynomial, must be inclosed in a parenthesis 
preceded by the sign — , or the sign of each of its terms must be changed. 



§ 259.] FRACTIONS. 187 

Case IV. 

258. Fractions reduced to their lowest common denominator. 

Fractions with unlike denominators may be reduced to equiv- 
alent fractions with a common denominator, as shown below. 

1. Reduce J, |, and ^ to equivalent fractions with their 
lowest common denominator. 

The L. C. M. of 4, 8, and 12 is 24 (§ 234) . Change the fractions to 24ths. 

• '• if » if » ^^^ if *^® ^^® equivalent fractions required. 

2. Reduce , "!" ,^ , and --^ to equivalent fractions 

2a ' 3a* ' 4:a^ ^ 

with their lowest common denominator. 
The L. C. M. of 2 a, Sa^, and 4a8 is 12 a». 

3a;-6 _ 6a2(3x-6) ^ 18q'^-36a8 . 
2a 12a8 12a8 ' 

6 + 5a; _ 4a(6 + 6g) ._. 24a + 20fla; . 
3a2 12a8 12a8 ' 

3x 3x3x 9aj 



4a8 12a8 12a* 

18 a%B — 36 a^ 24 a 4- 20 oflj j 9 aj ^v • i 4. *-^ *• 

. x%?u,^ — !fiiiL ^•'tt -r ^v/i*^ gj^^ _^j«/ j^yg ^jjg equivalent fractions 

12 a* 12 a8 12 a« 

required. 

3. Reduce -, -, and - to equivalent fractions with the 
be a 

lowest common denominator. 

The L. C. M. of b, c, and a is ahc, 

a a^c b ab^ c 6c» 

6 ~ abc * c ~ abc * a ~ abc 

.'. -Z-, -=-, and -T- are the equivalent fractions required. 
abc abc abc 

269. It is seen from the foregoing solution, that, when the 
denominators of the several fractions are prime to each other, 
they are reduced to like fractions by multiplying both terms 
of each fraction by aU the denomitiators except Us own. 



138 ALGEBRA. [§ 260. 

4. Eeduce ^"^ , ^~ . and ^ ^ ^ to equivalent fractions 
a—ba + b ar^b^ 

with the lowest common denominator. 

The L. C. M. of a - 6, a + 6, and a" - &» is a* - 6*. 

g + & _ (g +6)(a + &) _ (g + ft)^ . 
a - 6 g2 - 62 a^-b^ ' 

a-b _ (a-b)Ca'-b) _ (a- b)^ , 
a + b g2-62 a^-l^' 

3g 3g 



aa _ fe2 a2 _ 52 

•'• o o » o~  o » ^"^^ -T—^^ are the equivalent fractions required. 
g2 — 62 a* — 6^ (j2 _ 52 

Reduce to equivalent fractions with the lowest common 
denominator 

- a 6 1 c -^11,1 

5. — , — , and —. 10. -, -- — -, and — — -. 

be ac ab « — lar-— 1 a?-|-l 

3c4ac 6a a-\-bcr — b^a^b 

7. IT-, :r— r, and ----• 12. 



bx 2x^ lOa? ' a? + 2' a: + 3' aJ' + Sic + e 

8. «^, «Jb^, and ;^,. 13.^-^^-^ ^ 



3a' 4a2' 6a» * aj-3' aj-f3' iB«-9 

9. ?^±^, 5L±6 and ^^±^. 14. ~^^ ^ , ^-^ 
ajy aj*<^ a?y* o^ — y^ 7?-\-ip oc^ — y^ 



260. To reduce several fractions to equivalent fractions 
with the lowest common denominator. 

Find the L. 0. M, of the denominators of the several frdctioiis. 

Divide the L. C. M. found by the denominator of each fraction, 
and multiply both terms of the fraction by the quotient (§ 245) ; 
or multiply both terms of each fraction by those factors of the 
L. C. M. which are not contained in its denominator. 



§ 262.] FRACTIONS. 139 



ADDITION AND SUBTRACTION OF FRACTIONS. 

261. In arithmetic only like numbers can he added or sub- 
tracted; and hence, to add or subtract 5 yards and 2 feet, the 
yards must first be reduced to feet, or the feet to the fraction 
of a yard; and, to add or subtract f and |, both fractions 
must be changed to twelfths or to other like fractional unit. 

262. The same fundamental principle holds true in algebra; 
and hence, to add or subtract algebraic fractions, they must 
first be reduced to like fractions, i.e., to equivalent fractions 
with a common denominator. 

1. What is the sum and the difference of |^ and ^? Of 
}a and fa? 

The L. C. M. of 8 and 12 is 24. Hence 

f - A = if - M = A» difference. 
The L. 0. M. of 3 and 5 is 16. Hence 

2a, 3a 10a, 9a 19a ^^19^ ^^« ,4a „,^ 
= = , or — a, or a H , sum. 

3 6 15 16 15 ' 16 * 16 ' 



3a? --4 
8a 



2a _ 3a ^ 10a _ 9a ^ jflL, difference. 
3 6 15 16 16 

2. What is (1) the sum and (2) the difference of 

andif±^? 
12 a 

The L. C. M. of 8 a and 12 a is 24 a. Hence 

m ^3g--4 4a;+3 _ 3(3x-4)+2(4a;+3) _ 9a;~124-8x+6 _ 17fl;-6^ 
^ ^ 8a 12a 24a 24a 24a 

(2") 3g-4 4a;-|-3 _ 3C3a;-4)-2(4x+3) _ 9x-12-8a;~6 _ g-18 
^ ^ 8a 12a 24a 24a 24a 

It must be observed that when a fraction is preceded by the 
sign — , as in (2), the entire numerator is to be subtracted, and 
hence the sign of each term must be changed or be conceived 
to be changed. If the numerator is inclosed in a parenthesis, 
as above, the signs of the included terms must be changed 
when the parenthesis is removed (§ 104). 



140 ALG£BRA. [§ 263. 

263. The sum or the difference of algebraic fractions may 
be found by connecting them with the proper sign, and then 
reducing the result to its simplest form (§ 115). 

Simplify 

' 5x X ax 0/3^ ^(j?Q^ 

. 7 ,2 a ^ X 2y-x 2f-icy'\-o^ 

2x X or y ao aby 

g 0^ + 5 a? + 3 ^ 3?y-^f 3a;^4-3y» Gg'-a^ 

xy y * 5 a* 5icV 10 2^ 

6. 2+-. 10. — -^ — ^H ^ 

y a; X y ^ xy 

264. To add or subtract algebraic fractions, 

Reduce the given fractions to equivalent fractions with the 
lowest common denominator. 

Write the sum or the difference of the numerators of these 
fractions, as the case may be, over their lowest common denomr 
inator, and reduce the result to its simplest form, 

-, Q. Tij a — 2 , a — 1 a — 3 

11. Simplify -H -• 

^ ^ a + 1 a + 3 a'\-2 

The L. C. M. is (a + !)(« + 3) (a +2). 

(a - 2)(a + 3)(a + 2) = (a2 - 4)(a + 3) = «» + 3 a« - 4 a - 12 ; 

(a-l)(a + l)(a + 2) = (a2_i)(a + 2)=o»4-2a2_a_2; 

(a - 3)(a + 3)(a + 1) = -(a2 - 9)(a + !) = -(«« + a^ - 9o - 9). 

a'* + 3 a^ — 4 a — 12, 1st numerator ; 
Hence a' + 2 a^ — a — 2, 2d numerator ; 
— (a' + a^ _ 9 Of _ 9)^ 3(1 numerator. 

a^ + 4 a'-^ 4- 4 a ~ 5, sum of numerators. 
. fl-2 g-l q-3 ^ gS 4. 4^2 _^ 4q _ 5 
'a+1 a + 3 a4-2 (a + l)(a + 3)(a + 2)' 

Since the third fraction is preceded by the sign — , the sign of each 
term in its new numerator is conceived to be changed when combining 
the three numerators. 



§264.] 



FRACTIONS. 



141 



Simplify 



12. -J-+ 1 



13. 



x-\-y x — y 

1 1 

a — b a-\-b 



14. ^dbl+^izl. 

» — y x-\-y 
15. ^^^_H- "^ 



16. 



a^ + V a^-b^ 

-1 ^. 

x-3 aj-2 



17. ^+ 2 



18. 



x-\-4: x — 3 

lH-2a l-2a 
l-2a l4-2a' 



19 ^ — 1 I ^ — 3 a; + l 

a;-2 aj2_4 ^^2* 

ar — 2 aj-f4 



20. 



21. 



aj + 3 a — 5 

a -{-bx a — bx 
a — bx a-\-bx 



22. -1-+ ^' 



a + b a^-b^ 



23. 



24. 



25. 



a 



a^ 



a — 6 a^ — b^ 
1 



1 2a? 



aj — a x-\-a^ x^ — a^ 

a + b , a-b d' + b^ 
a—b a + b a* — 6* 



2B !+«» I l-iB' 2(l+a^) 






a — a; a + a; g? — o^ 
28. -1^+ ^ ^^ 



aj + y ^ — y ar^ — 3^ 



29. 



2a 



1 3a2 



a:^+aaj-fa^ a?— a o^—a^ 



30. -1-+ /~^„4 "^ 



31. 



a+6 a^^ab+b^ a^+b^ 
1 a^ 

   » -  • 



32. . ^ . + ^ 



33. 



34. 



a{x — y) x(y — x) 

a X 

x(a — b) a(a + b) 

a 5 a 60 



a — 6 o? — b^ a-\-b 



36. ^- + ^^^ * 



aj + y 0? — y^ ^-\-y^ 
36. _3_+^ + ^^ ^ 



37. 



38. 



2a-4 8-2a2 a4-2 
a? 1 , aj + 1 



a; — 1 0^ — X 
a + b . 2a 



+ 



X 



a' 



a — b ab + b^ 



39. 2±b_^_a-b_2(^+^^ 
a — b a + b a^ — b^ 

.^ m^ ran , n 

40.  \- • 

(m + nf {m+nf m+n 
41. ^-.— ^+ 2a, 



« - 3 aj + 3 (aj + 3)^ 



142 ALGEBRA. [§ 265. 

42. -J- + -^ ^±^ ^^. 

x — S x-{-5 ar* + 2a:-15 x + 5 

Ao 1 1 6a , gg + Sa 



a? — 3a x-^3a a^ — 9a^ x — 3a 
44. .-J_+ 1^,+ «^-l 



1 , 4aj 5x 

45. h 



3 



46. 



x-\-a {x + ay 0^ — 0? 

1 1 

aJi^jx+12 a^+aj-12' 



47. -r -;-4- 



aj2_aj-20 iB*-8aj4-16 aj2_9a._|.20 
48. ^^ + ^ 5 + ^-^ 



49. 



aja-5aj+6 aj2-8a; + 15 a^-7i»+10 
1 1 



oi^-\-{a'\-b)X'{'ab a^ +{a + c)a; + ac 

266. When advantageous, the signs of both terms of a frac- 
tion may be changed (§ 247), or the signs of an eaen number 
of factors in either term, without changing the sign before the 
fraction. The sign of an odd number of factors in either term 
may be changed if the sign before the fraction be also changed. 

Thus, since (6 - a) = - (a - 6), ^ -=r -^ -, 

(b — a)(c — a) (a — 6)(a — c) 

, a? —X 

and 



(6 — a)(a — c) (a — b)(a — c) 

50. Simphfy ^^ __ ^^^^ _ ^^ + ^^ _ ^^g^ _ ^^ + ^^ __ ^^^^ _ ^y 

112 
Change to , and — to 

^ (6-a)(6-c) (a-6)(6-c)' (c-a)(c-6) 

2 ., giving -i- J- -+ ^ 



(a — c)(6 - c) (a ~ 6)(a — c) (a — &)(6 — c) (^a-c)(b—c) 

The L. C. M. of the new denominators is (a — 6) (a — c)(6 — c). 
. 6--c--(a-c)+2(a-6) q-& 1 



(a - 6)Ca - c)(6 - c) (a - &)(a - c)(6 - c) (a - c)(6 - c) 



§ 267.] FRACTIONS. 148 

Simplify 

»- a X Ko c^ 5a 6b 

61. -: r: r' 52. 



53. 



x(a — b) a(b — a) a — b 6 — a a + b 

1 1 1^ 

(a + by b^ - a* (a-b)' 



s 



54. -r-^, r+ ^ 



{a-h){a-c) {b-a)(b-c) 

66. , 4 TT+ ^^ 



(x — a)(a —b) (a — x)(b — a) 

66. ^ + ^ + ^ 

(x-y)(x-z) (j/'-x){y-z) (z-x){z-y) 

a^-b , a-^b^ ab^l 

57. -: rr-z ZT "i 



(a-6)(a-l) (b-a)(b + l) (o-l)(6 + l) 

68. '^-y' + y*-^ 4- ^ 

(x+y)(x — z) (y+z)(x + y) (x — z)(y + z) 

69. 5±5 + ^±^ + 5±^^ . 

a?—(a-\-b)x-{-ab a^—{a-{-c)x-\-ac Qi^—(b+c)x-{-bc 



MULTIPLICATION AND DIVISION OF FRACTIONS. 



266. Since 2xn = — , and ±xn = -, it follows that a 

b b bn b 

fraction may be multiplied by mvMiplying its numerator or by 
dividing its denominator, 

267. Since ^^n = ^, and 55 ^ n = -^, it follows that a 

b b b bn 

fraction may be divided by dividing its numerator or by multi- 
plying its denominator, 

1. Multiply ^ by 6; — by m; — by xy. 

b my y 

2. Multiply — — by ajy; by Aa^; by aW. 

if 



144 ALGEBRA. [§ 268. 

3. Divide^ by 3; ^by3; ^ by ay. 

4. Divide ^^^ by 4a^ by Sa^y; by 4aY 

6. Multiply 4^8 by a- ft; bya + 6; bya«-6^ 
6. Divide 5^^-^ by a - 6 ; by a + 6; by a* -61 



Multiplication op Fractions. 

1. Multiply - by -. 

h n 

Since -^ = m-i-n, -x-- = -xm-4-n = =^-i-n==^» 
n n b b bh 

a m _ am 

b n bn 

3. Multiply 1^ (1) by J^; (2) by ^. 

oa — oo a 4- a-|-6 

^2N ax + &x gg _ ^ _ x(a + 6)(a2 - b^) _ (a + &)g 
^'^3a-36 a + 6 3(a-6)(a + 6) 3 

268. It is thus seen tbat the product of two algebraic frac- 
tions is the product of their numerators divided by the product of 
their denominators. 

Multiply 

4. 1^ by ^. 6. g("' - y) by ^0^ . 
5Qcy Say 5a^ 4:(x — y) 

2a!y -^ 6oV oft -^ as + y 



§ 271.] FRACTIONS. 145 

8. "LzJ^ by ^ + ^. 11. ^"^ by ?L±^. 

9. -^ by ?^^. 12. ^Il2!by— ^±? 

10. $ZJ^ by ^±i^. 13. (^ + yy by ^^. 

«^ + 2r « — y (« — y) « + y 

269. To multiply a fraction by a fraction, 

Multiply the numerators together for the required numerator^ 
and the denomin^ators for the required denominator, and reduce 
the resulting fraction to its lowest terms. 

Division of Fractions 

270. The reciprocal of a fraction is 1 divided by the fraction 
(§ 33). Thus, the reciprocal of f is 1-^|=|, and the reciprocal 

of - is 1 -^ - = — Hence the reciprocal of a fraction is the 
b b a 

fraction with its terms inverted. 

271. Since —= m-!-ri, ?-s-—=-^-(m-*-n): and since ? -s- m 

n b n b b 

is n times too small, --$-(m-*-n) = [--hm)xn = — xn = — • 

b \b J bm bm 

It is thus seen that ^ is divided by — by multiplying ^ by 

b n b 

the reciprocaJ> of — ; i.e., by — . 

n m 

1. Divide ^^ by 2" 



ab a — b 

ah ' a-b ab 2 a 2a%' 



2. Divide ~ by ^^ 



xy Q^ 

3 . 12 _ 3 ^ a;^ _ 3 osV _ ay 
xy ' a^ xy 12 12 xy 4* 

WHITB*S ALa. — 10 



146 ALGEBRA. [§ 272. 

3. Divide ?^^^ by ?^=±. 

-. -i = — X — '—- = a 4- o. 

a+6 a+6 a+6 a—h 

Divide 

4. 3a^ by i^ 9. (» + y)' by ^ + y 
10^^ 5a!j^ » — y (p — y)* 

6 6 a'b' (b — c)* 6' — c* 

6. ^ by ^('"-y) . 11. '^+y' by ^±y 

7. ^(f-y) by ^i:^. 12. ^"^-^^ by £l^. 

Sajy xy arH-ic + l a? + l 

8. -^ by ^"^^ 13. ^'-^' by -^i^H^. 

272. To divide a fraction by a fraction, 

Multiply the dividend by the reciprocal of the divisor; or nnuU 
tiply the dividend by the divisor with its terms inverted. 



EXBRCISBS m MULTIPLICATION AND DIVISION OP FRACTIONS. 

Simplify 

- ^ ax A: by ^ 3(a + &)^ 46 a — 

2b 9a* ^ 

„ Sa^6a^b^21b(^ 
« Sx ^ 5a 



4. 



2 7 
3a^ag , 6aa^ 
T7' 21f' 



5^^ 2UdYz 
' 3a^y^ Aa^b*(^' 



V. 


2 '^a + 6 * 6 


7. 


a — X m? ^ a 


a? a'-x'' a*-a? 


8. 


X ^ a ^a»-< 
a-{- X a — x Of 


9. 


a^^b^ ^^a^-b" ^^Sa 
a — b^ c? 5 


10. 


6{a^-f) _ 16(0! + 2,) 
y x — y 



§ 272.] FRACTIONS. 147 

jj SOCa" - a^ ^ 25(a - a;) ^^ a^-y* ^ »* 

a* — «* a + 05 aj(a? + y) ' aj(aj* — 2/*) 

a^ + ar a + x ar + x — 2 x-\-2 

13. ,?^^?^±^. 18. A+«Yl_2V 6_. 

15 <^ + a' . g^y 20 ^-^ . ^ + 2^ 

21. /q^a^ . g'-gy4-2A ^ g'4-a?y + y' _ 

' a^ + 3ic-88 a52 + 2aj-35 a^-{-2x-S 

fa?^9x-\-lS , a?-Sx-{-15 \ a^-Ux-\-40 
* Va52-17a?4-70 'aj«-4aj-2V a5«-3a?-18' 

a^4-20a?4-96 a^-8a;-20 . a^ + 10g4-16 
' aj*-16ajH-50 a^ + lOaj-24 * oj^-Taj + lO' 

3^4- 17 a? 4- 60 3^4- a; -56 . a^-H3a?4-40 
'*^- aj2_i4aj 4.49 aj2 + 5aj-. 84 * a?-7 

^ + y^ . /^ ^ — ay + y' y g Y 

a?(a-a;) a(a4-g) 1 

^^' a^ + 2ax-\-a'^a?-2ax-^a'^cuc 

( df^-¥ a^^V\ f a^h a-^b\ 
' \a^-V a? + v)\a-b a-\-bJ 

P* — 2pq 4- g* i>^ 4- 2pg 4- g* ' P^ 4- ^^ 

30. ^ + y' x "^"^^ . a^ + 2rcV4-y* 
' a^ ^y^ 01^ — a?y^ 4- 2^ ' a?* — 2/* 



28 



29. 



148 ALGEBRA. [§ 273. 

COMPLEX FRACTIONS. 

273. The division of an integer by a fraction, or a fraction 
by an integer, or a fraction by a fraction, may be expressed 
by writing the dividend over the divisor, with a line between 
them. 

Thus, 3 -s- 1 may be written ^ ; I -^ 3, | ; and | -^ f , I- 

274. A fraction which has a fraction in one or both of its 
terms is called a complex fraction. 

275. A complex fraction may be reduced to a simple frac- 
tion by performing the division indicated. 

1353^8 24 _, I 272^2 4 
- = — ! — = — X — = — ; and -^- — — *- — — — ^ - — — 

15 8 5 5 25' 3^ 



m. t353^8 24 ,^272^2 4 
Thus, •^ = --5-- = - x- = — ; and -2- = — -5-- = -x- = — • 
'458 66 25' 3i 3 2 37 21 



276. A complex fraction which contains only a simple frac- 
tion in one or both terms is best simplified by multiplying both 
terms by the L, 0. M, of their denominators, 

i i 

1. Reduce to a simple fraction (1) ^ ; (2) ^« 

2 + ?5 

b 2 

2. Reduce to a simple fraction (1) -j (2) 1 — =-• 

2a-- 1+- 

a a 

_A _ & _a(2b+a) 



a a 



,0. - 2 _.. fxq _.. 2a _ q + l-2a _ l— o 



§ 276.] FRACTIONS. 149 

a g — 6 

3. Reduce to a simple fraction — =■• 

b a — h 

g o + & ~ a(a-6)+6(q + &) ^ 6Ca2_2,3N aCa^ - 62)+ 6(a + 6)« 
6 0-6 6(a-6) ^ '^ 

gg - a62 -(ggft - 2g62 + 6») _ a8 - g^ft + ah^ - b^ 

~ g8 - ab'^ + g26 + 2 gfta + 68 - gS + g^^ ^. aft2 ^ 58 

Simplify 



a; 
6. 



1 



6. 



"g 


+ & 




1 


I* . 


1 


•*/ 


a 




c 


a 
a 


a& 


a 



a; 4-1 
14-- J--r-- — !-— 

X a __ 1 — a 



13. 



l-fa 



1-J a-i 1-a 
1. a 

aJ-4 -f . 1 14. 



^ -1 



. 


1 


1+ 






a 




a? 


X — 


• — 




a 



^ a 1 

a-^-b 



7. iZl^ t. ^ 16. 



oj — a 



a? XT i ^~^ 



x + a 



1 1 J I «-"! 



16. -^ j— 18. —J 

6-16+1 a+1 



1-. ^^ ^ a? 



1-a 



a + 2f 



160 



ALGEBRA. 



t§ 276. 



mSCBLLAIIBOnS BXBRGISRS. 



Eeduce to simplest form 
1. (^±^. 



2. 



3. 



a»-l 
(a - 1/ 

g*-l' 



4. (^^ - ^)^ 



5. 



6. 



g*-l 
g«-l' 

g*-a^ 



(g3-ar^(g + a) 

7 (g4-6y-(2g-f6)« 
g + &-(2g4-&) 



8. 



9. 



ij^ + x -20 
a^-x-12 

2aj*H-5aj-12 



4a?-9 



10. g — 



g' 



11. 



a — b 
c^ — a? 



12. -^--1- ^ 



«-l 



aj(a? — 1) 



-o g , 2g 2g* 

lO. r "T" 



14. 



g — 1 g + 1 g* — 1 

X , ___g« 



05 — g a^ — a^ 



15. -i--2+ 1 



16. 



17. 



g— 1 g g+1 

a?-h5 ga?4-2 

"y ay 

x-\-2y x — 2y 
x — 2y X'\-2y 



18. -A^+ S"' 6 



a? + l aj* — 1 a5 — 1 



19. 



X 



f 



x — y a? — f a* + ^ 

20. gj^- %"^X 
4 g 5(g + a?) 

„- g — aj.g-haj cf — a? 



go; 



OJ" 



4gV 



22. ^-^±^'x^Il^. 
ajy ojy xy 

23. g — ^ ^ "^ ^ X ^ 



a + b a — b a-^-b 

24 a^ + 3^^ 1 x-^ 

a? — 3 aj + 3 aj + 3 

25. A+^Ufl-^^Y 

V g + V V g + V 
26. -T^F^xfg-l^Y 

^ / , 2aj \ / 2aj \ 
V x-Sj \ x-Sj 



g-f 6 a? — y 



§ 276.] FRACTIONS. 161 

29 a^-Sy* 3a?*4-3y* 6a?-^xy' ^ 

30. « 2y-3a? ^ 23/'-4a?y + g' 
y ab aby 

31. y — g a? — ga^ + yg* + afe ^ 

32. ^ + y^ . a^^ + a^y-^a^ + 2r* ^ 
« — 2^ ' xy 

\l + m my\l + m 1 — my 

35. (— ^^ + -H--— ^- i 

se /^^ -\-n m — n\ /m + w wi — n\ 
\m — n m-\-nJ ' \m — n m -h n/ 

37. / "c^ + ft a-6\ /g + fe I g — & Y 
\g — 6 a-^-bJ \a — b a-\-bJ 

38 / a^ + y  a?-y \ / a?4-y a?-y\ 
\g + 6 "^ g — 6y * \^g — 6 g + 6/ 

39. f^^^^±lVr^?^-^^\ 
\ X y J \ X y ) 

40. / "^ I '^ \ ,( '^ I ^\ 
\^m ■\-n m — nj' \m 4- w m — nJ 

41. Z'-!!?^ n_Vr-^^+— ^\ 

\m — n m-\-nJ \m + n m — nj 

42 ^* — a?* . a^ -\-Qfi cf — Q^ , g^ — gV + a:!* 

g*-faJ* g«-aJ« g^ + o^ * g* + gV + aJ*' 

43. / ^ ^ I ^ V ^ ^ 

\l4-g 1 — aJ'l — a l + g 

44 /" a? ^ \^( y _ *^ 



152 ALGEBRA. [§ 277. 



CHAPTER X. 
SIMPLE EQUATIONS CONTAINING iPRACTIONS. 

277. An equation may be cleared of fractions hy multiplying 
both members by the L. C, M. of the denominators (§ 141, II.). 

In practice it will often be found convenient to transpose 
the terms, and combine those not fractional, before clearing 
the equation of fractions, as shown below. 

1. Solve the equation 2aj — |aj + 5 = |aj + 12. 

Transposing and combining, 2x — fa; — |a; = 7; 

multiplying by 12 (L. C. M.), 24x - 8x - 9x = 84 ; 

combining terms, 7 x = 84 ; 

whence x = 12. 

2. Solve the equation ^±1-^Z1^_7 = 6-^. 

^ 3 5 10 

Transposing and combining, ^Jt x — 6 _j_ _« _ ^3 . 

clearing of fractions, 10 x + 10 - (6 x - 18) + 9 x = 390 ; 

transposing and combining, 13 x = 362 ; 

whence x = 27|J. 

When the numerator of a fraction preceded by the sign — 
contains more than one term, it will be found convenient tc 
write the new numerator in a parenthesis, as above, and then 
consider the signs changed when combining the terms. 

3. Solve the equation — L_ = ^±^. 

The L. C. M . of the denominators is x* — 9. 

Clearing equation of fractions, 5x + 15— (x — 3) = x + 30; 

transposing and combining, 3 x = 12 ; 

whence x = 4. 



§278.] SIMPLE EQUATIONS. 168 

278. When the denominators are partly monomial and partly 
polynomial, it may be found advantageous to remove the mono- 
mial denominator before the polynomial. 

4. Solve the equation ^ - liSl^l^ = 3 - i^. 

^ 5 x + 1 10 

Multiplying both members by 10, 4 a: - ^C^ ~ ^) = 30 - 1 + 4 x ; 

x+ 1 

uraiisi>osing and combining, ^ — \^~ ) = 29 

x+1 

clearing of fractions, - 30x + 60 = 29x + 29; 

transposing and combining, — 59x = — 31 ; 

multiplying by — 1, 59 x = 31.; 

whence ^ = H* 

It may sometimes be convenient to combine the fractions in each mem- 
ber of the equation before clearing of fractions. 

Solve tlie equations 

5. ^x — ^x = 5. 15. a; -»- 1(11 — a;) = ^(19 — a?). 

6. 20.-5 = ^^-1. 16. 4 ^ 



4 8 aj-2 aj-4 

7. x-{-ix-{-ix=:ll. 3 2 



18 A+?=ii 

«  «_-«o « 2a! a; 12 



19. a!-26zi£ = ^iai 



*• 2 + 3-^^ 4 

10. a; + ^-5 = 4a!-17. ^ » 

2 3 

11. 5_£±i = a!-3. ^**' 3ac~6^2»* 

,-. 2»+3 a;+3 a!-4 , „ 21- ^^-(a!-7)=3. 

12. —3 — = -^ + 3. x-3 

13. 50,-^+12 = ^+26. 22. -_^ = ^-^. 

^A ie-5 . fi^ 284 -a? _„ 12a; + 97 _ 4a; + 16 

14. ___+6x = — ^— . 23. ^^-p^- -^-2- 



154 ALGEBRA. [§ 278. 



24 



. ^^-gf+^ = 3» + 10. 29. -^+-1 



x — 2 a^—1 x-\-l 1 — x 



25. |-4 = |-^ 30. 3 - 2 _ 1 ^0. 

3aj-7 3a?-5 1-a? l + a? 1-aj* 



a!_6 4_a! ai»-9 a; + 3 3-x 

27. £±l-£ri3^1 82. 1 1 - ^ ^ 



a5 — 1 a5 + 3 oj x—2 05—4 a5— 6 a— 8 

28 g(«^-2) 2(a;-3) ^3 33 '^-l , »+ l_ 2(a>-2) 
a! + 2 aj + 3 ' x-2'^a! + 2~ iB + 2 

84. ^-^ + 10 = ^. 

35. Sx-\(2x + 6)=-l-i{llx-32). 

36. 2 - ^(6aj - 4) =i(4a;- 18)- a. 

37 a? +4 •^^ a?-23 a?-l 
4 5 7 * 

38. __-3a. + 4 = - ^— . 

39. i(7a? + 5)-|(16 + 4aj) + 6 = i(3a? + 9). 

40. |(3a; + 4)-|-i(16-a?)+i(3-7aj) = 0. 

., 17 — 3aj 4a;+2 ^ ^^.7. ,o>, 

41. ^ = o — 6x-\'-(x + 2). 

o o o 



42. 



3a? + 2 2aj-4 3-2a? 



43 a?4-l a? __ 9 — jc .g- 8 



03 — 1 aj — 2 7 — a? G — x 
^ x'-x + l^a^ + x + l^^^ 

x—1 «+ 1 

46. 2a?~ + a^=5a?-2a?. 

(3a;-2)(2a?- 3) 6a? -8 2^ . g+lO. 
6 5 "^15 3 



1 280.] 



SIMPLE EQUATIONS. 



155 



LITERAL EQUATIONS. 

279. The known numbers in an equation may be represented 
by letters, the first letters of the alphabet being commonly used 
for this purpose (§ 11). Thus, in the equation ax-]-bx = ab, 
a and b may represent known numbers ; i.e., numbers regarded 
as known. 

280. An equation in which some or all of the known numbers 
axe represented by letters is called a literal equation. 



1. Solve the equation 



a 



X — b X — a 



Clearing of fractions, 
transposing terms, . 
factoring, 
dividing by a — 6, 



flwc — a^ = 6a! — 6^ ; 
ax — hx = a^ — h^', 
(a - h)x = (a — 6)(a + 6); 
x = o + 6. 



V X 

2. Solve the equation - + b = -+cL 

€L C 



Transposing terms, 

clearing of fractions, 
factoring, 

dividing by c — o, 



a c 
ex — ax== acd — abc ; 
(c — a)x = ac(d — 6) ; 
ac(d — h) 

X = ' 

c — a 



Solve 

3. ax+c = bx + 2c. 

4. ^ax + ^bx = c. 

- CLX ^ mn , J 
5. c = [-a, 

b n 

^ a? 05 — 6 d 

b. — ^ ^ —• 



a 



7. 



X 



X 



a+b a-rb 



= -1. 



8. — \-n = b. 

m a 

9. - — ^x + c= ^ x — cL 
a + b a — b 

m — x x-^r 



10. 



x+p 



n 



X 



-, a^^ax b^ + bx ^ 
11. — ; — ! — = a?. 

b a 



156 ALGEBRA. [§ 280. 



ax 





X — 


a 


x-^a 7? — 


•a' 


15. 


2+ 

X 


b__ 

X 


d ax + b 
X a? 




16. 


£'- 


-ab 


= 1 + 6. 





12. 2£zi^ + 65JlA' = a.. 17. o-ft^o + J 

b a x-b x + b a^-b^ 

,^ cLX — b aX'\-b 5 ,« / , v/ .,v , .., 

^^- -^Ts "^33=^39- '®- (p + <^)ip + ^) = {^ + cy. 

a 6 6 1^- {x—a)(x-b) = {x—a—b)\ 

14. ^ — = -j J. 

20. m*aj+2mn— n^=m^+w'. 

21. 6w^a?— 2m^=3mn— 4mna?. 

22. - + - + -=i)^+jpr + gr. 
a? a p q r 

Probibms. 

1. Divide the number 132 into two parts such that one part 
may be ^ of the other. 

2. Divide the number a into two parts such that one part 
may be — of the other. 

3. Divide the number 108 into three parts such that \ of 
the first equals \ of the second, and also \ of the third. 

4. Divide the number n into three parts such that - of the 

11 ^ 

first, - of the second, and - of the third, will be equal. 

b c 

Let X = first part. 

Then ? x 6 = — = second part, 

a a 

and 2 X c = — = third part. 

a a 

Hence aj + — + - = n. 

a a 

Solving the equation, x = — — , first part. 

a-\-b -\- c 

Whence — = — ^ — , second part ; 

a a -\-b -\- c 

— = — — — , third part. 
a a-hb-hc 



§280.] SIMPLE EQUATIONS. 157 

5. Divide the number m into three parts such that - of the 

11 ^ 

first, -— of the second, and -— of the third, will be equal. 

2a oa 

6. A man walked 87 miles in 3 days, and ^ of the distance 
walked the first day equaled \ of the distance walked the 
second day, and ^ of the distance walked the second day 
equaled ^ the distance walked the third day. How far did 
he walk each day? 

7. A father bequeathed ^ of his estate to his eldest son, 
■^ of the remainder to his second son, and the rest to his 
youngest son; and the eldest son received $1200 more than 
the younger. What was the share of each? 

8. Divide $735 among three persons so that the second 
will have f as much as the first, and the third ^ as much as 
the other two together. 

9. A man bought a horse and carriage for $ 275, and ^ of 
the cost of the carriage plus $ 33 was equal to J of the cost of 
the horse. What was the cost of each ? 

10. A man bought a horse, saddle, and bridle for $150. 
The cost of the saddle was ^ of the cost of the horse, and the 
cost of the bridle was ^ the cost of the saddle. What was the 
cost of each ? 

11. Ten years ago A's age was f of B's age, and 10 years 
hence A's age will be f of B's age. What is the age of each 
now? 

12. At the time of marriage a wife's age was f of the age of 
her husband, and 12 years after marriage her age was ^ of her 
husband's age. How old was each at marriage ? 

13. A man can do ^ of a piece of work in a day, and a boy 
can do \ of it in a day. In how many days can both of them 
working together do it ? 



168 ALGEBRA. [§ 280. 

14. A can do a piece of work in a days, and 6 in & days. In 
how many days can both together do it ? 

Let X = number of days. 

Then ~ = part both can do in one day ; 

X 

^ = part A can do in one day ; 
a 

- = part B can do in one day. 
b 

Hence 1 + 1 = 1 

a b X 

Clearing of fractions, bx-{- ax=:ab; 

whence x = , number of days. 

a-\-b 

15. A can do a piece of work in 8 days, and B in 12 days. 
In how many days can both together do it ? 

16. A can do a piece of work in 3 days, B in 5 days, and C 
in 6 days. In how many days can they together do it ? 

17. A and B can do a piece of work in 8 days, A and C in 
10 days, and B and C in 12 days. In how many days can A, 
B, and C together do it ? In how many days can each alone 
doit? 

18. A and B working together can build a wall in 8 days, 
and A alone can build it in 12 days. In how many days can B 
alone build it ? 

19. A and B can build a fence in 8 days, and with C's help 
they can build it in 6 days. How long will it take C alone to 
build the fence ? 

20. A man spent J of his money, and then earned ^ as much 
as he had spent, and then had $ 21 less than he had at first. 
How much money did he have at first ? 

21. An estate was so divided between two heirs that J of 
the share of the first equaled f of the share of the second ; and 
the difference of their shares was $ 362. What was the share 
of each ? 



§ 280.] SIMPLE EQUATIONS. 169 

22. A man paid $ 8100 for two farms, and f of the cost of 
the larger farm was equal to -^ of the cost of the smaller. 
What was the cost of each ? 

23. A piece of carpeting containing 135 yards was cut into 
three carpets such that ^ of the number of yards in the first 
carpet equaled i of the number of yards in the second, and f of 
the number of yards in the third. How many yards were in 
each carpet? 

24. A cistern can be filled in 4 hours by two pipes running 
together, and in 6\ hours by one pipe alone. In how many 
hours can the other pipe alone fill it ? 

25. Two stoves consumed a certain amount of coal in 12 
days, and the smaller stove would consume it in 30 days. In 
how many days would the larger stove consume it ? 

26. A farmer paid $ 410 for sheep of different grades. For 
■J- of the whole number he paid $ 10 each ; for ^ of the whole, 
$ 7.50 each ; and for the rest, $ 5 each. How many sheep did 
he buy ? 

27. The sum of $100.50 is contributed by 100 persons. 
Some give 50^ each ; ^ as many, 75^ each ; and the rest, $ 1.50 
each. How many contributors of each class ? 

28. A and B have equal incomes. A lays up ^ of his each 
year ; but B spends ^ more than A, and in 3 years finds him- 
self $ 600 in debt. What is the income of each ? 

29. The perimeter of a triangular field is 81 rods, and the 
longest side is ^ longer than one of the other sides, and twice 
as long as the other. What is the length of each side? 

80. A woman sold from her basket \ the number of eggs in 
it, and then sold ^ of the eggs remaining, and then had 20 
eggs left. How many eggs did she sell ? 



160 ALGEBRA. [§ 280. 

31. The difference of the squares of two consecutive num- 
bers is 21. What are the numbers ? 

32. A factory employs 200 men, 150 women, and 80 chil- 
dren, and they receive each week as wages $4110, 3 men 
receiving as much as 4 women or 8 children. How much did 
each man, woman, and child receive per week ? 

33. Divide $ 6000 among A, B, and C, giving A f 300 more 
than B, and C one half as much as A and B together. 

34. Divide m dollars among A, B, and C, giving A n dollars 
more than B, and n dollars less than C. 

35. A merchant bought a lot of velvet at $ 1.50 a yard, and 
then sold one half of it at $ 2 a yard, one third of it at 
$ 1.75 a yard, and the remainder at $ 1.25 a yard, and made a 
profit of $ 157.50. How many yards did he buy ? 

38. Divide m dollars between A and B in the ratio of a 
to b. 

Let ax = A^s share, 

and hx = B's share. 

Then ax + bx = m; 

whence x= ^ 



a-hb 



Hence ox = -«^, A's share; 

Ix = -5^, B's share. 

a + b 

37. Divide $ 120 between two men in the ratio of 3 to 5. 

Suggestion. This problem may be solved by substituting f$ 120 for 
m, 3 for a, and 6 for 6, in the foregoing formulas, for A*s and B*s shares. 

38. The difference between two numbers is 25, and the 
greater is to the less as 3 to 2. What are the numbers ? 

39. The difference between two numbers is m, and the 
greater is to the less as a to 6. What are the numbers ? 



§280.] SIMPLE EQUATIONS. 161 

40. A boy bought apples at 2 for a cent, and as many more 
at 3 for a cent, and then sold them at the rate of 6 for 3 cents, 
and gained 11 cents. How many apples did he buy ? 

41. A fruit vender bought a certain number of pears at 

2 cents each, ^ as many lemons at 3 cents each, and ^ as 
many oranges at 4 cents each, and paid $1.96 for the lot. 
How many of each did he buy ? 

42. A party of 20 persons pay $ 40 for their railroad tickets. 
The full fare is $ 2.50, but the children are charged only half 
fare. How many children in the party ? 

43. A miller made a mixture of barley, com, and oats, using 

3 bushels of barley to 4 of corn and 5 of oats. How many 
bushels of each grain did he use in a mixture of 72 bushels ? 

44. When a colonel tried to draw up his regiment in a solid 
square with a certain number of men in the front rank, he had 
35 men too many, and when he put one man more in the front 
rank he had 30 men too few. How many men in the regi- 
ment ? 

Suggestion. Let x = the number of men in the front rank of the first 
solid square. 

45. A woman bought a dollar's worth of postage stamps, 
receiving a certain number of five-cent stamps, twice as many 
two-cent stamps less 3, and 3 times as many one-cent stamps 
less 2. How many stamps of each kind did she buy ? 

46. A speculator bought a piece of land at $ 450 an acre, 
reserved 5 acres for himself, and laid out the rest in building 
lots which he sold at $ 1000 an acre, gaining in the transaction 
$11,500 besides the 5 acres reserved. How many acres of 
land did he buy ? 

47. A man engaged to work for 30 days on the conditions 
that he was to receive $1.50 for each day he worked, and 
forfeit 50^ for each idle day. At the end of the 30 days he 
received $ 27. How many days had he worked ? 

WHITENS ALO. — 11 



162 ALGEBRA. [§ 280. 

48. A man engaged to work for m days on the conditions 
that he was to receive a dollars for each day he worked, and 
forfeit b dollars for each idle day. At the end of m days he 
received n dollars. How many days had he worked ? 

49. A farmer can mow a field in 12 hours, his oldest son in 
16 hours, and his second son in 18 hours. In how many hours 
can the three together mow it ? 

50. A, B, and C can do a piece of work in 20 days ; A and 
B can do it in 40 days; and A and C in 30 days. In how 
many days can each alone do it ? 

51. A man being asked his age replied that f of his age 10 
years ago is equal to f of his age 10 years hence. What was 
his age ? 

52. At what time between 2 and 3 o'clock are the hands of 
a watch together ? At what time between 4 and 5 ? 

53. At what time between 3 and 4 o'clock are the hands 
of a watch opposite each other? At what time between 8 
and 9 o'clock ? 

54. A man spends one fifth of his yearly income for house 
rent, one half of the remainder for provisions, two fifths of the 
remainder for other expenses, and lays up $ 240. What is his 
yearly income ? 

55. A steamer, running 18 miles an hour, follows a ship 16 
miles off, that is sailing 10 miles an hour. How many miles 
must the steamer run to overtake the ship ? 

56. A courier starts from a certain place and travels at the 
rate of 10 1 miles an hour. Two hours later a second courier, 
traveling 13 1 miles an hour, is sent to recall the first. In how 
many hours will the second courier overtake the first ? 

57. A person has just 5 hours at his disposal. How far 
can he ride with a friend in a buggy, going 10 miles an hour, 
and walk back at the rate of 4 miles an hour ? 



§282.] SIMPLE EQUATIONS. 163 



GENERAL PROBLEMS. 

281. When the given numbers in a problem are represented 
by letters, it is called a general problem ; and its solution is a 
general solution for all problems of that class. It also serves 
as a model solution. Several of the foregoing problems are 
general problems. 

282. The result obtained by the solution of a general prob- 
lem is called a solution formula, or, briefly, a formula ; and, by 
substituting for the letters in such formula the particular num- 
bers given in a similar problem, the numerical answer to such 
problem is obtained. 

1. The time past noon is n times the time to midnight. 
What is the time of day? 

Let X = time past noon ; 

then 12 — X = time to midnight. 

Hence « = n (12 — x). 

Transposing terms, x-\-nx=l2n; 

12 n 
whence x = — —, time past noon ; (1) 

1 -f w 

12 - X = 12 - i?-^ = -^^, time to midnight. (2) 
1 + w 1 + n 

2. The time past noon is f of the time to midnight. What 
is the time of day ? 

Let x = time past noon, and substitute | for n in Formulas (1) and 
(2) above, and we have 

X = i^ = ^iii = 3d = 44 = 4h. 48 m., tune past noon. 
1 + n I i 

12 - aj = -i?- = — = 7i = 7 h. 12 m., time to midnight. 

3. A man, being asked the time of day, said that f of the 
time past noon equals | of the time to midnight. What was 
the hour of day ? 



164 ALGEBRA. [§ 282. 

4. What is the time of day when |^ of the time to noon is 
equal to ■§• of the time past midnight ? 

6. A courier pursues a second courier, who has a start of m 
miles ; and the first courier travels at the rate of a miles an 
hour, and the second at the rate of b miles an hour. In how 
many hours will the first courier overtake the second ? 

Let X = the number of hours ; 

then ax = distance traveled by first courier, 

and bx = distance traveled by second courier^ 

Hence ax—bx = m; 

whence x = ^* , number of hours. 

a — b 

6. An express train running 36 miles an hour follows an 
accommodation train running 24 miles an hour, and the accom- 
modation train has a start of 84 miles. In how many hours 
will the express train overtake the accommodation train ? 

Let X = number of hours. 

Since « = — ^L_, and m = 84, a = 36, and 5 = 24, 

a — b 

84 

X = = 7, number of hours. 

36-24 

7. A train sets out from A for B, which is n miles distant, 
running a miles an hour ; c hours later another train leaves B 
for A, running b miles an hour. How far will each train have 
run when it meets the other ? 

Let X = the number of hours run by first train ; 

then ax = distance run by first train ; 

X — c = number of hours run by second train ; 
b(x — c) = distance run by second train. 
Hence ax + b(x — c)=n; 

whence x = ^-±^ ; 

a + b 

ax = "(^ "^ ^^) , distance run by first train ; (1) 
o + 6 

6(x — c) = ^t^.Il-^£2, distance run by second train. (2) 



§282.] SIMPLE EQUATIONS. 166 

8. The railroad distance from Buffalo to Chicago is 540 
miles. An express train leaves Buffalo for Chicago, running 
at the rate of 35 miles an hour ; and 5^ hours later an express 
train leaves Chicago for Buffalo, running at the rate of 40 miles 
an hour. How far will each train have run when they meet ? 

SuGGESTioy. Solve this problem by the use of the general formulas 
(1) and (2) obtained in solving Problem 7. 

9. Divide the number m into four parts such that the first 
increased by w, the second diminished by n, the third multi- 
plied by w, and the fourth divided by w, will be equal. 

Let X = number to which the results are equal ; 

then x — n — first part ; 

jc + n = second part ; 

- = third part ; 
n 

nx = fourth part. , 

Hence a!-n + « + n + ?+ na! = «. 

n 

Combining terms, 2a54--4- wx = wi; 

n 

clearing of fractions, 2nx-\-x-\- n^x = mn ; 

factoring, (»^ + 2 w + l)a! = mn ; 

whence x = ^^ 



(n + 1)2 

35 — n = — ^^ ^ — n, first part ; 
(n + 1)2 

x+n = — ^^^ 1- n, second part ; 

(n + 1)2 

* ^ -. third part ; 



n (n + 1)2 

nx = , ^^ — -, fourth part. 
(n + l)2 

10. Divide the number 80 into four parts such that the first 
increased by 3, the second diminished by 3, the third multi- 
plied by 3, and the fourth divided by 3, will all be equal. 

SuoGEBTiON. Let x denote the number to which the several results 
are equal, x^S denoting the first part, a; + 3 the second part, and so on ; 
or solve by substituting in the foregoing formulas. 



166 ALGEBRA. [§ 283. 

11. Divide the number 100 into three parts such that the 
first increased by 8, the second diminished by 8, and the third 
divided by 3, will be equal. 

12. Divide the number m into three parts such that the 
first increased by n, the second diminished by n, and the third 
divided by n, will be equal. 

13. The sum of two numbers is m, and their difference is 
n. What are the numbers ? 

Let X = the larger number ; 

then X — n = the smaller number. 

Hence x-\-x — n = m. 

Transposing terms, 2 as = w + n ; 

x=™±»,thelargernnmben 0) 

X - n = ^^^-±-^ - n = ^Lm^, smaller number. (2) 

2 2 ^ ^ 

283. It is seen from these two formulas that when the sum 
and the difference of two numbers are given, 

I. To find the larger number, add the sum and the difference 
of the two numbers^ and divide the resuU by 2, 

II. To find the smaller number, subtract the difference of 
the two numbers from their sum, and divide the result by 2, 

284. In like manner the formula obtained by the solution 
of any general problem may be expanded into a rule for the 
solution of all like problems. 

Such a formula may also be used for the solution of problems that vaiy 
somewhat from the general problem from which it is derived. 

14. Expand into a rule the formula reached by the solution 
of Problem 1. 

15. Expand into a rule the formula reached in the solution 
of Problem 5. 



§288.] SIMPLE EQUATIONS. 167 

285. Percentage formulas. — The problems in simple percent- 
age involve three numbers such that, if two are given, the third 
may be found. These numbers may be represented by letters, 
as follows : 

h = base, or the number of which the per cent is taken, 
r = rate per cent, or the number of hundredths. 
p =z percentage, or the number found by taking the given per 
cent of the base. 

Since p = hr, r =^j and 6 = — , and hence the formulas : 

r 

(1) p=br. (2) r = |. (3) 6 = |. (4) 6=^- 

By these formulas all the problems in simple percentage 
may be solved. 

16. If 5% of a certain ore is silver, how much silver is 
there in 3740 pounds of the ore? 

17. 2400 pounds of iron ore yielded 1008 pounds of iron. 
What per cent of the ore was iron ? 

18. If copper ore is 25% copper, how many pounds of the 
ore will yield 1000 pounds of copper ? 

286. Interest formulas. — The problems in interest involve 
five numbers or elements : the principal, the rate, the time, the 
interest, and the amount. These five numbers are so related, 
that, if any three are given, the other two may be found. 

287. These numbers may be represented by letters, as fol- 
lows: 

p = principal, or the money on which interest is computed. 
r = rate per cent, or the number of hundredths. 
t = time, expressed in years or parts of a year. 
i = interest, or the money paid for the use of the principal. 
a = amount, or the sum of principal and interest. 

288. The interest on any sum of money for one year is 
found by multiplying the principal by the rate per cent, ex- 



168 ALGEBRA. [§ 289. 

pressed as hundredths; and the interest for any given time 

is found by multiplying the interest for one year by the 

number of years. Hence i=prt. ' 

• • • 

Since i = prt, r = — -, t= — , and p = —; and we thus obtain 

pr pr rt 

the following special formulas for the solution of the several 

classes of problems in interest : 

(1) i^prt, (2) r=±. (3) t = ±^. (4) p = l 

19. What is the interest of $ 160.80 for 2 yr. 3 mo. at 8 % ? 

20. The interest of $ 95.40 for 3 yr. 9 mo. is f 28.62. What 
is the rate per cent ? 

21. The interest of $56.78 for a certain time at 10 % was 
$ 22.24. What was the time ? 

22. What principal will produce $86.80 of interest in 2yr. 
4 mo. at 6% ? 

289. Since the amount is the sum of principal and interest 
(a = j9 -h t), a=p -^-ptv] and hence, factoring, a =i)(l 4- rt), 

and p=- Again, since a=p-\-ptr, a—p=:ptr\ and 

hence r = -, and t = —- 

pt pr 

We thus obtain the following formulas : 

(1) a=p + i=p+prt. (3) < = ^- 

a—p /A\ ^ — ^. *• ^ 



(2) r = ^^^. (4) p = a-t = 



pt ^ ' l-{-rt 

23. A note of $95.40 Sit S^oj when paid, amounted to 
124.02. How long did the note run ? 

24. A note bearing interest at 9 % amounted in 3 yr. 2 mo. 
to $ 360.75. What was the face of the note ? 

Note. For additional problems in percentage and interest, see White's 
'«New Complete Arithmetic,*' pp. 178-192 and 240-246. 



5 293.] SIMULTANEOUS EQUATIONS. 169 



CHAPTER XL 

SIMULTANEOUS EQUATIONS. 
SIMPLE EQUATIONS WITH TWO UNKNOWN NUMBERS. 

290. In a simple equation with only one unknown number, 
there is one value of this number, and only one, that will 
satisfy the equation. Such an equation is said to be determinate ; 
i.e., the unknown number in it has a definite value. Thus, (1) 
3 a; = 12, and (2) | a; = 10, are determinate equations, the value 
of X in (1) being 4, and in (2), 15. 

291. When an equation contains two unknown numbers, as 
X -{- y = 12, an indefinite number of values of these unknown 
numbers will satisfy the equation, and for this reason the 
equation is said to be indeterminate. Thus, in a; + y = 12, the 
value of X will vary with the value of y, and the value of y will 
vary with the value of oj. If ^ = 4, a; = 8 ; it y= 2, a? = 10 ; 
if aj = 7, y = 5i if x = W, 2/ = — 3; and so on. 

292. Two equations that express different relations between 
two unknown numbers are said to be independent. Thus, 
x-\-y = 12, and ocy = 35, express different relations between x 
and y, and hence are independent equations. 

Independent equations cannot be made to assume the same form. 
Thus, x + y = 12 (1), and 2 a; + 2 2/ = 24 (2), can be made to assume the 
same form by multiplying each member of (1) by 2, or dividing each 
member of (2) by 2, and hence these equations are not independent. 

293. The unknown numbers in two independent equations 
may have the same value in both ; and, when this is the case, 
the substitution of these values for the unknown numbers will 
satisfy both equations simultaneously. Thus, the values of x 



170 ALGEBRA. [§ 294. 

and y in a; — y = 2, and xy = 35, are respectively 7 and 6, and 
the substitution of these values satisfies both equations. 

294. Two independent equations in which the two unknown 
numbers have each the same value are called simultaneous 
equations. Thus, a; + 3/ = 13, and a; — ^ = 3, are simultaneous 
equations. 

Elimination. 

295. The solution of two simultaneous equations involves 
their combination in such maimer as to remove or eliminate 
one of the unknown numbers. 

Thus, let 2 a; + 1/ = 42, and 3 a? — 2^ = 33, be two simultaneous 
equations. Adding the first members and the second mem- 
bers of the two equations (Ax. 1), we have 

2x + y = 42 (1) 

3a?-y = 33 (2) 

5 a; =75 .-. a? = 15. 

Substituting 15 for x in (1), we have 30 + ^ = 42. .-. y = 12. 

296. The process of combining two simultaneous equations 
in such manner as to obtain a single equation with only one 
unknown number, is called elimination. 

297. There are three general methods of elimination : to wit, 

I. By addition or subtraction. 
II. By substitution, 
III. By comparison 



298. I. Elimination by addition or subtraction. 

1 . Find the values of x and y in the equations 

5a; + 4y = 40, (1) 
7x-2y = lS. (2) 

Multiplying both members of (2) by 2, 14x — 4 y = 36 ; (3) 

adding (1) and (3) member to member, 19a; = 76 ; 

whence x = 4. 



{ 



§298.] SMlXTAXEOrS EQCATrOX?. 


in 


SnlEtifcmiag^iarziiL :i ^ 


21)^4^=40; 




tnm^msing, etc. 


4j=a>; 




whence 


J = ^ 




Hence 2=4, and y=ri; and tboe vaioBB of jr and y w€l aalas^ tj^ 


giren equatiaBS. 






2. Solve tbe eqoatioiis 






Midtiplyiiig (1) bj 2, and (2) br 3» 


8i-u6j = ll-\ 


v3^ 




15x-h6jr = ia>; 


K*^ 


sabtiactiiig (3) from (4), 


7x = 70; 




whence 


x = 10L 




Sahsthnting 10 lor x in (1\ 


4O + 3y = o0; 




tnuisposing; 


3jr = lo; 




whence 


f = a 






M=^ 


0^ 


3. Solve the eqnatiann 


5 ' 3 


v-n 


Clearing (1) and (2) of fractions, 


5x + 4> = U!iX 


vSX 




9x + 10y-S30; 


v«^ 


multiplying (3) by 6, and (4) by 2, 


25i + 20y = !»0. 


V''^ 




18i + 20y = tS»iO; 


v«»^ 


subtracting (6) from (5), 


7* = l-»0; 




whence 


a! = 20. 




Substituting 20 for x in (3), 


100 + 4y = U50; 




transposing, 


4y = t50; 




whence 


J»=15. 





It is sometimes not necessary to clear the equations of fractions, since 
the coefficients of one of the letters may otherwise be made the same, as 
shown in the following solution. 



i 



172 



ALGEBRA. 



[§ 298. 



4. Solve the equations 



Multiplying (1) by 2, 

adding (2) and (3), 
whence 

Substituting 8 for x in (1), 

transposing, 

Solve the equations 

( x + y = lS, 
^' |2aj-y = 21. 

raj4-2y = 20, 

(4:x--3y = 13, 
^' (6aj-4y = 22. 

®' l5x-2y = SS. 



(x-h3y = 36, 
Xx-2y = 16. 

1 7a; -32^ = 62. 

(3x + 4.y = U, 
^^' \^y_Sx= 1. 



9. 



10. 



12. 



3a; + ^ = 28, 

5aj-^ = 37. 
2 



6a 2 
8 6 



= 6, 



4 3 



££_? = 4. 



5x^ 
4 3 

§^ or2x 



x = 



«+l 



12; 

16; 

8. 
6; 



(1) 
(2) 



(3) 



V 
6 



? = 1. /. ysA. 



IS, -^ 






= 9, 



«_3/ 
3 6 



^-^ = 1. 



14. < 



( 2x 
5 
3x 

I 8 



32 
4 

3y 



15. ^ 



T + 6 



x.y 
8 6 



= 6, 
= 6|. 

14, 
4. 



16. ^ 



3a; n — = Si« 



§299.] SIMULTANEOUS EQUATIONS. 173 

To eliminate by addition or subtraction, 

Multiply or divide the given equations by such numbers as wiU 
make the coefficients of one of the unknown numbers equal in 
the resulting equations. 

If the equal coefficients have unlike signs, add the resulting 
equations; and, if they have like signs, subtract one equation from 
the other. 

Note. The adding of the first members and the second members of 
two equations is called briefly the adding of the equations; and in like 
manner the subtracting of the first and the second members of one equa- 
tion respectively from the first and the second members of another equa- 
tion is called briefly the subtracting of one equation from the other. 

299. II. Elimination by substitution. 

17. Solve the equations \j^^V^^'^^' 9^ 

^ \2x-3y= 5. .(2) 



Eliminate x. 






From (1) we find 




a; = 20-2y. 


Substituting 20 - 2 y for » in 


(2), 


2(20 -2y)- 32/ = 5; 


simplifying, 




40-7y = 5; 


transposing, 




-7y = -35; 


dividing by — 7, 




y = 5. 


Substituting 5 for ^ in (1), 




a; + 10 = 20 ; 


whence 




X = 10. 


18. Solve the equations 




(6y-3aj = -l, (1) 
l3» + 42/ = 37. (2) 


Eliminate y. 






From (1) we find 




,.5^ 


Substituting 5*^ for y in 

5 


(2). 


5 


clearing of fractions, etc.. 




27a; = 189; 


whence 




« = 7. 


Substituting 7 for x in (1), 




6y-21=-l; 


whence 




y = 4. 



174 



ALGEBRA. 



[§300. 



Solve the equations 



19. 



20. 



21. 



22. 



23. 



24. 



i2x-^3y = 22, 

j2x + 3y = 27, 
(4aj— y= 5. 

(2a; + 5^^ = 19, 
( x-7y= 0. 

(5a? + 22/ = 29, 
\2y- x = -l. 

( y 

(5a; 



25. 



f4a; + 3y = 
\2y-4cx = 



22, 
-12. 



26. 



I 



33.-- 
4 



= 10, 



= a 



27. < 



22!=-6, 

2^4-22 = 34. 

5a;-32^ = 34, 
3a;= 2. 



28. 



2^3 ' 

- + ^ = 74 

2 "^ 2 
2a;-32/ = 0. 



= 6, 



To eliminate by substitution, 

Find from one of the given equations the value of the unknoum 
number to be eliminated. 

Substitute this value for the unTaiown number in the other 
equation, and solve the resulting equation, 

300. III. Elimination by comparison. 

29. Find the values of x and y in 



{ 



Eliminate x. 
From (1) we find 

and from (2), 



solving the equation, 

Substituting 6 for y in (1), 
whence 



a; + 2t/ = 25, 


(1) 


2x-3y = lb. 


(2) 


x = 25-2y; 


(3) 


^ 16+3y 
*- 2 • 


(4) 


(3) and (4), 26-2y=i^-^; 




y = 6. 




X + 10 = 26; 




« = 16. 





§300. J 



SIMULTANEOUS EQUATIONS. 



175 



30. Solve the equations 

Eliminate y. 
From (1) we find 

and from (2), 

Equating the values of y, 

solving the equation, 

Substituting 18 for z in (1), 

whence 



4 + 3~^' 


(1) 


y 2 ., 

.3 6~ 


(2) 


V = 36-^; 




, = 3+1 




8+1 = 36-^; 




a = 18. 




1+6 = 9; 




tf = 12. 





Solve the equations 



31. 



32. 



33. 



34. 



35 



36. 



37 



38. 



i2x- y 

X + 2Z: 

2y-Sz: 
4:y — 5z: 

5 X-{-4:Z. 

ilx-^z 



{ 
{ 



{ 
{ 
{ 
{ 



y- 25 = 

3^ + 52 = 

4:X + 12y 

X+ 6y 

y-2z = 
4:y-Sz = 

x + Sz = 
3x-'2z = 



= 16, 
= 36. 

= 18, 
= 8. 

= -9. 

= 11, 
= 15. 

= 4, 
= 20. 

= 5, 

= 2. 

-6, 
26. 

50, 

7. 



39. 



40. 



41. 



42. 



43. 



I 



i+^=>* 



^ y ^ 

X—Z = 4:y 

Sx , 2z K 

T+3- = ^- 

L~3 4 ''*• 



{ 



ax — by = 10, 
ax-\-by = 26. 

4 + T"^^' 

^2 T"" -^^ 



176 ALGEBRA. [§301. 

To eliminate by comparison, 

Find from each of the given equaiiona the value of one of the 
uiiknown numbers in terms of the other. 

Equate the values of the unknovm number thus founds ana 
solve the resulting equxUion. 

MISCBLLAIVBOUS EXBRCISBS. 

301. The following pairs of simultaneous equations may be 
solved by any one of the above methods of elimination. The 
forms of the given equations will usually indicate which 
method will be the most advantageous. 

« 

302. It will sometimes be best to simplify the equations 
before determining the method of elimination to be used. 

Thus, the equations 

(1) ^_?LzJ^ = 2,and(2)^+^ = 10, 

may first be changed (1) to x-{-lly = 60, and (2) to 
5 a? + 2/ = 60, and then either x or y may be eliminated by 
subtraction or by substitution. 

The method by addition or subtraction will usually be found the most 
advantageous ; but, when the coeificient of one of the unknown numbers 
in either equation is unity, the method by substitution may be preferable. 



Solve the equations 



^ (x + y = 19, ^ <llx + 3y = 

\x^y=: 7, '(4a? — 7y = 

2 <2x-^Sy = lS, g <15x-13y = 

\5x + 4ty = 22, ' (17a?- y = 



= 100, 
4. 



= 21, 
65. 



3 ( 7a?+ 42/ = 85, (Sx-5y= 5, 

(I3aj-37y = 69. ' (5a;-3y = 



3y = 36. 



§302.] 



7. 



8. 



9. 



10. 



SmULTANfiOUS EQUATIONS. 



177 



11. 



12. 



31. 



Xl9x-21y = - 

<25x^21y= 410, 

Xl6x-14:y = -12. 

14:X-13y= 51, 

152^— a; = 123. 

x-j-2y = 3, 
5y-{-4LX = 6. 



13. 



14. 



I 
I 

J 



15. 



16. 



2 3 



5 + 2 = 8. 
13 2 






17. 



18. 



f60a;-17y = 146, 
(48a; + 13y = 170. 

f4x-f-12y = 5, 
( aj+ 2y = l. 

f ia:-f 3y = 8, 
(K3a:-2)=2y + 3. 

2aj-y= 7. 



7 3 
a? — y = 4. 



19. 



20. 



21. 



22. 



ii(^ + y)-K^-y)= 2, 
U(«^ + y)+i(x-2,)=lo. 

((» + 5)(2^ + 7)=(« + l)(y-9)+112, 
I 2a; + 10 = 3v + l. 



23 

24 

25 

26 
27 






(5f2^-lla; = 4y + 117|, 
I 8a; + 175 = 2y. 

f 49a? - 37f + 4Sy = 9x + 273, 
X 3x--26{ = -5y. 



13a;4-72^ = 121, 
2x + \y=. 14. 

10^a;-5|2/ = 40, 
9a; -8.52^ = 20. 

.2a; + .3y = 4.3, 
.2 a? -.2 2^ = 1.5. 

1.7 a; - 2.3 y = 1.6, 

.4 a; + .06 y = 2.18. 

.2 a; 4- .4 2^ = 2.8, 

.4 a; + .3 2^ = 3.1. 
whitb's alo. — 12 



28. < 



29. 



1 



4 + 4 ^t' 

x + 2 y + 2 _.. 
6 6 ~ 

a 



a; 



30. 



|a;-f-2^=», 
la? — v = d. 



IT 



«. 






— -.-- = «- 

» ft 



?^ = « — 






--^r=<» 



33. 






X 

n 



* 

I 

m 






37. 






2, 
X 



34. 



f-^ 2_= 1 

)a-rh a — h ^ 

1 a-j- ^ a — h 






X 



303. When simultaneons equations are fractional, and of the 

form -±-= Cy they may be solved most readily by treating — 
X y 

and - as the unknown numbers. 

y 

\ X t/ 

89* Solve the equations < ^^ ^ 



Kliminate z. 
Multiplying (1) by 8, 

multiplying (2) by 2, 

Nubtracting (4) from (8), 

Hubitituting 8 for y in (2), 



^ + ?l = 18; 

» y 



?1 = 3. 






(1) 

(2) 
(3) 



§304.] 



SIMULTANEOUS EQUATIONS, 



179 



Solve the equations 



40. 



? + ? = !, 

X y 

^ 2J y 



41. < 



42. < 



1* 3_3 
X y ^ 

ay 
rl , 1__ 8 
X y JlO 
112 



a; 



2^ 15' 



43. < 



cm , ?i 

— + - = «, 



- + — = 6. 

La? y 



45. 



46. 



44. < 



X y 



5 

6' 



2_3 



5 
6 



7 
a; 



y 



47. < 



^aj y 

X y 

n m 

X y 



a? 2/ 

a? 2/ 



1 
6^ 
23 
21 ' 

= P, 



EQUATIONS WITH THREE OR MORE UNKNOWN NUMBERS. 

304. When three equations with three unknown numbers 
are given, we may combine one of the equations with each of 
the other two in such manner as to obtain two equations with 
two unknown numbers. The two resulting equations may then 
be combined, as in the preceding cases, giving a single equa- 
tion with one unknown number. The value of this unknown 
number being found, the value of the other unknown numbers 
may be obtained by substitution. This process may some- 
times be shortened, as shown in Example 2. 



1. Find the values of x, y, and z in 

3aj + 52^+ 2 = 26, 
6a; + 3^ + 32 = 36, 
9x + 4:y + 4:Z = 50. 



(1) 
(2) 
(3) 



180 ALGEBRA. [§ 305. 

First eliminate z. 

Multiplying (1) by 3, 9 x + 16 y + 3 a = 78 ; (4) 

subtracting (2) from (4) , 3 x + 12 y = 42. (5) 

Multiplying (1) by 4, 12 x + 20 y + 4 5? = 104 ; (6) 

su^t^acting (3) from (6), 3 x + 16 y = 64. (7) 

Next eliminate x from (5) and (7). 

Subtracting (5) from (7), 4 y = 12. .-. y = 3. 

Substituting 3 for y in (5), 3 x = 6. .•. x = 2. 

Substituting 3 for y, and 2 for x, in (1), « = 6. 

Hence x = 2, y = 3, « = 5. 

r2aj + 32/+ « = 25, (1) 

2. Solve the equations -^ 4 a; + 5 1/ + 2 « = 46, (2) 

(60? + 42^ + 42; = 58. (3) 

Multiplying (1) by 2, 4 x + 6 y + 2 « = 60 ; (4) 

subtracting (2) from (4), y = 4. 

Multiplying (1) by 3, 6 x + 9 y + 3 « = 76 ; (6) 

subtracting (3) from (5), 6 y — « = 17 ; (6) 

substituting 4 for y in (6), z = 3. 

Substituting 4 for y, and 3 for «, in (1), 2 x = 10. . •. x = 5. 

305. When any one of the unknown numbers does not occur 
in all three equations, we may first eliminate such unknown 
number from the equations in which it does occur, thus obtain- 
ing two equations with two unknown numbers. 



(x-\-y = 


11, 




(1) 


3. Solve the equations \x-\-% = 


:10, 




(2) 


(y + z = 


9. 




(3) 


First eliminate x from (1) and (2) 








Subtracting (2) from (1), 




y — a = l. 


W 


Adding (4) and (3), 




2y = 10. 


.•. y = 6. 


Substituting 6 for y in (3), 




« = 4. 




Substituting 6 for y in (1), 




x = 6. 





These equations may also be solved by first adding (1), (2), and (3), 
then dividing the resulting equation by 2, and from this equation sub- 
tracting successively (1), C2), and (3). 



§306.] SIMULTANEOUS EQUATIONS. 181 

306. When four or more equations with four or more un- 
known numbers are given, one of the unknown numbers may 
be eliminated by combining one of the equations in which it 
occurs with each other equation in which it occurs ; then a 
second unknown nunxber may be eliminated by combining one 
of the resulting equations in which it joccurs with each other 
resulting equation in which it occurs ; and so on. 

When each imknown number does not occur in all of the 
given equations, the process may be shortened. There must 
be as many given equations as there are unknown numbers. 



4. Solve the equations 



x + y-\-z= 9, (1) 

x-{-y — z-{'W=: 15, (2) 

x — y-\-z-]-to = ll, (3) 

Sx-2y + 4:Z= 9. (4) 

First eliminate w, 

(2)-(3), 2y-2« = 4. (5) 

Next eliminate x in (1) and (4). 

(I)x3, 3a; + 3y + 3« = 27; (6) 

(6) -(4), &y^z=lS, (7) 

(6)-2, y-z = 2; (8) 

(7)-(8), 4y = 16. ... y = 4. 

Substituting 4 for y in (8), z = 2. 

Substituting values of y and 2; in (1), x = 3. 

Substituting values of x, y, and 2; in (2), t(7 = 10. 

Solve the equations 

3aj + 5y — 2 = 18, r7a; + 3y — 52 = 3, 

5. -<{5a; — 6.y-f52;= 8, 8. ■] 3z — y — 4:X = l, 
054-^ + 2=6. ( X + y — Z=:l. 

4a; + 22/-42 = 22, r Sx-3y -\-4:Z = 25, 

6. ■{5x — 4:y-\-2z = 2Sy 9. } 5x-^2y — z= 9, 
5z-x + Sy==2S. (4y-llaj-f32; = ll. 

2x + y-\-6z = A6, r 5aj — 4y4-»=3, 

7. ■{6x-\-^y-Sz = 16, 10. -] 3aj-h6y-22; = 14, 
4cx-'6y + 4:z = 12. ( 4.x + 5y — z = lS. 



182 



ALGEBRA. 



[§306. 



11. S6aj + 4y-62; = 28, 
6y — z + 7x = 56. 

x — Sy'^2z= 0, 

12. •^3x + 2y--5z = 22, 
x — y — z= 2. 

13. 4x — y-\-z= 6, 
(« — y — « = — 2. 

i« + i3^ + i2 = 62, 

5x--Sy __-, 

42j-7a?" ' 



14. 



15. 



16. -< 



3z^x _^^ 

Sy-^x 

3^+_52 = i 

42 + 5 



17. 



« — y — 2 

y-f-x-^-z 



= 8, 

= 7, 



18. < 



19. < 



y — « — 2 

y — a?4-2 = — 2. 
a; y 2 4 

a 
« z 



1 1 



20. -< 



7aj-3y = l, 
ll2-7v = l, 

42-7y = l, 
19 a? - 3 V = 1. 



PROBLEMS INVOLVING SIMULTANEOUS EQUATIONS. 

Note. In the statement of a problem there must be as many equa- 
tions as there are letters representing unknown numbers (§ 306). 

1. The sum of two numbers is 343, and their difference is 
49. What are the numbers ? 

2. Divide the number 89 into two parts such that ^ of the 
greater part will exceed ^ of the less by unity. 

3. A boy, being asked his age and that of his sister, replied, 
" If I were 3 years older, I would be 3 times as old as my 
sister ; but, if she were 2 years older, she would be J as old as 
I am." How old was each ? 



§306.] SIMULTANEOUS EQUATIONS. 188 

4. If A were a years older than he is, he would be m times 
as old as £ ; but, if B were b years older than he is, he would 

be - the age of A. How old is each ? 
n 

6. A man has $ 10,000 in two investments. From the first 
he gets 6% interest; and from the second, 4% ; yet the second 
investment yields twice as much income as the first. What is 
the amount of each investment ? 

6. In an alloy of silver and copper, ^ of the whole, and 
42 ounces more, was silver ; while the copper was 8 ounces less 
than ^ of the whole. How many ounces were there of each ? 

7. In an alloy of nickel, copper, and silver, -J^'of the whole, 
plus 8 ounces, is nickel ; •§■ of the whole, plus 4 ounces, is cop- 
per ; and ^ of the whole is silver. How many ounces of each 
metal in the alloy ? 

8. A certain number expressed by two digits is equal to 
4 times the sum of those digits; but, if 18 be added to the 
number, the digits will be reversed. Find the number. 

9. Said A to B, " If you give me $ 100, 1 shall then have 
as much money as you." — " Nay," replied B, " give me $ 100, 
and then I shall have 3 times as much as you." How much 
money had each ? 

10. If 10 lb. of tea, with 35 lb. of sugar, cost $ 11.30, and 
12 lb. of tea with 25 lb. of sugar cost $ 12.20, what is the 
price of each per pound ? 

11. If a lb. of tea with b lb. of sugar cost m cents, but 
c lb. of tea with d lb. of sugar cost n cents, what is the cost 
of each per pound ? 

12. If the numerator of a certain fraction be multiplied by 
2, and its denominator increased by 2, the result will be equal 
to unity ; but if the denominator be multiplied by 2, and the 
numerator increased by 3, the result will be equal to ^. What 
is the fraction ? 



184 ALGEBRA. [§ 306. 

13. A certain fraction becomes equal to ^ when 3 is added 
to both its terms, but becomes equal to ^ when the same num- 
ber is taken from both its terms. Find the fraction. 

14. A certain fraction becomes equal to | if 2 be taken from 
its numerator, and it becomes equal to unity if 3 be taken from 
its denominator. Find the fraction. 

15. A certain fraction which is equal to f is increased to f 
by having the same number added to both its terms, and is 
multiplied by 2 by having another number taken from both 
its terms. Find the numbers. 

16. A fishing rod consists of two parts. The length of the 
upper part is to that of the lower part as 5 to 7 ; and 9 times 
the upper part, with 13 times the lower part, is 36 inches more 
than 11 times the length of the whole rod. Find the length 
of each part. 

17. A dealer has two sorts of tea. By mixing them at the 
rate of 3 lb. of the finer to 5 lb. of the coarser, he can sell 
the mixture at 95 cents a pound; but, mixing at the rate of 
1 lb. of the finer to 3 lb. of the coarser, he can sell the 
mixture at 90 cents a pound. What is the price per pound of 
each kind of tea ? 

18. The difference of two numbers is 8, and twice the sum 
of their reciprocals is equal to 3 times the difference of their 
reciprocals. Find the numbers. 

19. Seven years ago A's age was just 3 times that of B, 
but 7 years hence A's age will be just double that of B. 
What is the age of each? 

20. A person, wishing to give 25 cents each to a certain 
number of persons, found he had not enough money by 25 
cents ; but he could give each 23 cents and have 1 cent left. 
How many persons were there, and what sum had he ? 



§306.] SIMULTANEOUS EQUATIONS. 185 

21. A cistern can be filled in 6 hours by two pipes running 
together, and the one pipe fills as much of the cistern in J of 
an hour as the other does in 1 hour. In what time could liiey 
separately fill the cistern ? 

22. There are three numbers, such that the first together 
with I of the second is equal to 19 ; ^ of the second with f of 
the third is equal to 23 ; and ^ of the third with ^ of the first 
is equal to the second. What are these numbers ? 

23. A certain number has three digits whose sum is 15. 
The digit in the units place is 3 times that in the hundreds 
place ; and, if 396 be added to the number, the order of these 
digits will be reversed in the result. What is the number ? 

24. A and B together earn $ 50 in 8 days ; A and C together, 
$ 69 in 12 days ; B and C together, $ 55 in 10 days. How 
much can each earn in a day? 

25. A jeweler sold three rings. The price of the first with 
-J. that of the second and third was $25; the price of the 
second with ^ that of the first and third was $ 26 ; and the 
price of the third with ^ that of the first and second was 
$ 29. What was the price of each ? 

26. A person finds he can buy with $ 31.10 either 10 bushels 
of wheat, 12 of rye, and 9 of oats ; or 12 bushels of wheat, 
6 of rye, and 13 of oats ; or 16 bushels of wheat, 10 of rye, and 
2 of oats. What is the price of each grain per bushel ? 

27. There are four numbers such that, by adding each to 
twice the sum of the remaining three, we obtain 46, 43, 41, 
and 38 respectively. What are the numbers ? 

28. A man has two horses, and also a saddle worth $ 10. 
If he puts the saddle on the first horse, his value will be double 
that of the second horse; but, if he puts the saddle on the 
second horse, his value will be f 13 less than that of the first. 
What is the value of each horse ? 



186 ALGEBRA. [§307. 



CHAPTER XII. 

INVOLUTION AND EVOLUTION. 

POWERS. 

807. The power of a number is the product obtained by- 
taking the number one or more times as a factor (§ 27). The 
factor taken one or more times is called the base of the power. 

308. The exponent of a number denotes the degree of its 
power (§ 29). The exponent 2 denotes the second degree ; the 
exponent 3, the third degree; and the exponent n, the nth 
degree. 

309. A number that is composed entirely of equal factors 
is a perfect power; and a number that is not composed of 
equal factors is an imperfect power. 

310. Involution is the process of raising numbers to required 
powers. 

Since any power of a number is the continued product of the number 
by itself, involution involves the principles of multiplication. 

311. 25^=25x25; (Saf^SaxSaxSa-, and (3a)» = 
3a X Sa X Sa ••• ton factors. Hence 

Any power of a number may be found by taking the num- 
ber as many times as a factor as there are ernes in the eonponerU 
of the required power. 

1. What is the fourth power of 4 ? Of 5 ? Of 10 ? 

2. What is the third power of a ? Oix? Oiy? 



§315.] INVOLUTIOX AND EVOLUTION. 187 

312. (ay = a? xa^xa^ = a^^^ = a^^^ = a^ and (a^y = 
a"^ X a"* X oT ••• to n factors = a"^* = a*". Hence 

Any power of a number may be raised to any required power 
by multiplying the exponent of the given power by the exponent 
of the required power. 

3. What is the cube of a* ? Of a?*? Ofy*? 

4. What is the nth power of a* ? Of or'? Ofy*? 

5. What is the nth power of a- ? Of6»? Ofaf? 

Monomials. 

813. 123 = (3 X 4)» = 3» X 4«; (3a)» = 3« x a«; and (3a)» = 
3* X a". Hence 

Any power of a number is ^/le ^wodwc^ of each of the factors 
of the number raised to the required power, 

1. What is the third power of 2x5? Of 4x3? 

2. What is the fourth power of ooj* ? Of a^b^x? 

3. What is the fifth power of 2 ab^c ? Of 3 a^xf ? 

314. It is thus seen that any power of a monomial may be 
found by raising each of its factors to the required power, 

316. /'2Y = «x^ = 5^^ = ^;; /^2Y=2x^x2 = ^; and 
\b) b b b xb b^' \b) b b b 6»' 

( - ) = 2 X - X - ••• to w factors = — • Hence 
\b) b b b 6» 

A fraction is raised to any power by raising both its terms 
to the required power. 

4. What is the fourth power of -? Of ^? 

b ar 

6. What is the wth power of -? Of — ? 

7 3r 



188 ALGEBRA. [§ 316. 

316. (-fa)'=(+a)x(4-a)x(+a)=-ha'; and (+a)»= +a*. 
(-«)«=(- a) x(-a)=-|-a^ (- a)8=(+a»)x(- a) = - a»; 
(— aF) X (— a)= 4-a*; (+ a*) x (— a)= — a*; and so on. Hence 

I. If a number is positive, all its powers are positive. 
II. If a number is negative, its even powers are positive, 
and its odd powers negative. 

317. This principle may be expressed by the formulas 
(-|-a)" = -|-a*; (-a)*" = +a"*; and (- a)*"+^ = - a*»+i 

318. It is thus seen that all even powers of a number are 
positive, and all odd powers have the sign of the number itself 

It follows that —a\ —a*, etc., are not perfect powers. See ** Imagi- 
nary Numbers'' (§ 413). The term number in §§316, 318, denotes a 
real number ; i.e., a number that is not imaginary. 

6. What is the third power of - 3 d'h ? Of 4 ab^ ? 

7. What is the third power of |^? Of -|^? Of -^? 

2b 3x 3 aoi 



Raise to the indicated power 

8. (Sa^by. 17. (-' 2 a^icT^y. ^^ f -5xy \*^ 

9. (5aV)'- IS. (3 abx)"^. \-^^J 

10. (^4.ab'xy. 19. (^^xyz)^, ^5. (^^Y. 

11. (-a^bcx^\ 20. (^abc)^^\ ^ "^ ^ 

12. (3a35^c)«. ^^ 26. (^Y- 

13. {--2c^xf)\ ^ ^ ^ Va^y 

15. (ahhir ^^ /zl2^^ 28. ( -"T"' 

16. (arb'^cy. \ 5ab^ J \ «/ 

819. To raise a monomial to any power, 

Raise the numerical coefficient to the required power, multiply 
the exponent of each literal factor by the exponent which indicates 
the required power, and prefix the proper sign. 



§322.] INVOLUTION AND EVOLUTION. 189 

320. A fraction is raised to any power by raising both terms 
to the required power, and prefixing the proper sign. 

Binomials. 

321. It may be found by actual multiplication that 

(a + &)' = a^ + 2a6 + ^. 

(a -[- 6)3 = «» -h 3 a^h + 3 a&2 + &^ 

(a + ft) * = a' + 4 a^ft + 6 a^ft^ + 4 a5» + b\ 

(a^-bf = a'-\-ba'b^- 10a%^ + 10 a%8 + 6 a5* + ft*, 

{a-bf = a^-2ab^-b\ 

(a - bf = o^ -Sa^b -{-Sab^- b^ 

(a - by = a* -4:a^b + Ga^ft^ _ 4a6» + ft*. 

(a- ft)* = a'^ - 5a*ft + lOa^ft^ - 10 a^ft^ + 5 aft* - 6». 

322. An inspection of the above powers of a + ft and a — ft 
shows the following facts or laws : 

I. The number of terms in each power of the binomial is 
one more than the exponent of the power. 

II. The exponent of a in the first term is the same as the 
exponent of the power of the binomial, and it decreases by 1 
in each succeeding term. 

III. The exponent of ft in the second term is 1, and it increases 
by 1 in each succeeding term. 

b does DOt appear in the first term, nor a in the last term. 
The sum of the exponents of a and b in each term is the exponent of 
the power of the binomial. 

IV. The coefficient of the first term is 1, and the coefficient 
of the second term is the same as the exponent of the power 
of the binomial ; and generally the coefficient of any term is 
the product of the coefficient of the preceding term by the 
exponent of a in that term, divided by the number of that 
term. 



190 ALGEBRA. [§ 323. 

For example, in the fourth power of a-{-b, the coefficient of 
the third term is 4 x 3 -*- 2 = 6. 

The coefficients of the first, second, third, etc., terms from the left are 
respectively equal to the coefficients of the first, second, third, etc., terms 
from the right. 

V. When both a and b are positive, all the terms of the 
power are positive ; and when b is negative, the odd terms are 
positive, and the even terms are negative. 

323. It is to be noted that the above equations denoting the 
powers of a + b and a—b are identities, and are true for 
all values of a and 6. 

324. It will be shown later (Chapter XXI.) that the fore- 
going laws hold good for any power of a binomial indicated by 
a positive integral exponent. Hence, if n denotes any positive 
integer, 

/ . 7.\« « . n !». . w(n— 1) ^ 2,2 , n(n—l)(n—2) ^ ..^ , 

This identity is called the binomial formula. 

325. A binomial may be expanded to any power indicated 
by a positive integral exponent by the application of the above 
formula. 

1. Expand (x—yf. 

What is the number of terms in the expansion ? 

What is the exponent of x in the first term ? In the third 
term ? In the last term but one ? 

What is the exponent of y in the second term? In each 
sequent term ? 

What is the coefficient of the first term? Of the second 
term ? Of each sequent term ? 

How is the coefficient of the third term found? Of the 
fourth term ? 

What is the sign of each term ? 



§327.] INVOLUTION AND EVOLUTION. 191 

Expand in like manner 

2. (a-bf. 5. (a--xy. 8. (a + 6)* 

3. (m-ny. 6. (x + yf. 9. (1 - xf. 

4. (a-\-xy. 7. (c-d)*. 10. {x-iy. 

Write the fourth term of 
11. (a -by. 12. (c + d)^. 18. (x-yy. 

Write the first three terms of 
14. (a -by. 15. (c+d)« 16. {x-yy^. 

Write the last three terms of 
17. (m-ny. 18. (a + xy. 19. (a — by. 

326. When one or both terms of a binomial contain mord 
than one factor, as Sa—l^, the binomial may in like manner 
be raised to any power. 

20. What is the third power oi2iX? — Sy? 
Let a = 2a?, and b = Sy. 

Formula. (a - by = a?- 3a^b -^Sal^-V. 

Substituting 2 a? for a, and 3 y for b, 

(2a?-Syy=(2x^^-S(2x^\Sy) + S(2a?)(3yy-(Syy 

=:Safi-36a^y + 54aY -- 272/3. 

Note. This is in effect the same as the inclosing of each term con- 
taining more than one factor in a parenthesis, and treating it as a or b. 

327. Since each term, the first and last excepted, of a 
binomial power higher than the first power, is composed of 
three factors, the expansion of such a binomial power as 
(2a^ — Syy may be somewhat facilitated by writing these three 
factors in a column, and then forming their product as below. 

Formula. {a-by=a^ -Sa^b H-3a5« -V. 

(2a^' -3 +3 -(3yy 

(2a^2 2a^ 

3y (3yy 

(2x'-3yy = Sa^-36a^-h5^asy-27y^ 



ld!2 ALoeBttiL. [§ 328. 

Expand as above 

21. (3a-2by. 23. fab-^» 25. (2m -ny. 

22. (a«-36c)». 24. (^-^cV. 26- (^"^^J' 

Polynomials. 

328. Polynomials of three or four terms may in like man- 
ner be raised to any power. 

1. Wliat is the third power of 2 a' -h 5 — c ? 

2 a2 + 6 - c = 2a« + (6 - c). 

Let X = 2 a^ and y = b — c. 

Formula, (x + yy = afi + Sx^ + 80^2 + ^8 

Substituting, (2a2)» +3 +3 +(6-c)« 

(2a2)a 2a2 
(6-c) (6-c)2 

(2a2 4- 6 - c)8 = 8 06 + 12 a*6 - 12 a*c + 6a^b^ - I2a^bc 

+ 6a2c2+ 68 - 362c + 36c5« - c8. 

2. Expand (a + b-^-c — d)\ 

+ 6 + c — df=(a+&) + (c — d). 

Let a; = a + 6, and y = c — (?. 

Formula. (x + yy = i)fi + Sx^ + 3x2/2 + y». 

Substitutmg, (a + 6)8 +3 +3 + (c - d)« 

(a + 6)2 (a + 6) 
(c - d) (c - d)2 

(a + 6 + c - d) 8 = a* + 3a26 + 3a62 + 68 + 3a2c + 606c 
+ 362c - 3a2d - 6abd-Sb^d + 3ac2 - 6ac(l + 3a<P 
+,3 6c2 - 66cd+ 36^2 + c8 - 3c2d + 3cd2 - d8. 

329. Since the square of a polynomial is composed of (1) 
the square of each term, and (2) twice the product of each 
term multiplied in succession by the terms which follow it 
(§ 161), the square of a polynomial may be directly written. 

Thus, (a + 6 -f c)« = a» 4. 6^ 4. c2 + 2a(6 4- c) + 2 6 X c 

= a* + Z^^ + c» + 2 aft + 2 ac + 2 6c. 



§331.] INVOLUTION AND EVOLUTION. 19*3 

The cube of a trinomial is composed of (1) the cube of each term ; (2) 
three times the product of the square of each term multiplied by the sum 
of the other two terms ; and (3) six times the product of the three terms. 

Eaise to the indicated power 
4. (6a6Va*«)3. ' [my ' \ 2ab^cJ 

6. (-Sm^nx^yy. \ SnyJ \ xfz J 

7. {-fxf)\ ^^ /-2aW ^^ /^ 

8. {-da?yzy. '\2mrt'j' ' \ard 



Min,n\ m 



24. 



(f-^)" 



Expand by the binomial formula 

15. (m + w)*. 20. (1-2^*. 

16. {m-n)\ 21. (af^-3a»)^ 

17. (a«-&)». 22. (^-^zy. 25. (»-^J- 

1~ i' 23. (2a + f\ 26. (o^^-^- 

19. (l-aj)«. \^ 2y V «/ 

Write the first three and the last three terms of 
27. (x-\-y)^. 28. {x-ay\ 29. {x + y)^. 

Write the sixth sequent term of 
30. {a-cf. 31. (l-xy^ 32. (x-Vf. 

ROOTS. 

330. The root of a number is one of the equal factors which 
multiplied together will produce the number (§ 34). 

331. The second or square root of a number is one of its two 
equal factors; the third or cube root, one of its three equal 
factors ; and the nth root, one of its n equal factors. Hence 
the nth root of a** is a ; i.e., the root of any power is its base. 

white's alq. — 13 



194 ALGEBRA. [§ 332. 

332. The root of a number may be indicated by the radical 
sign, -y/. Thus, V25 or V25 denotes the square root of 2b \ 
\/64, the cube root of 64 ; and VaS, the nth root of db (§ 35). 

The figure or letter written in the opening of the radical 
sign is called the index of the root. 

The root of a number may also be indicated by a fractional exponerU* 
1 • 1 5 

Thus, a^ denotes the square root of a ; a", the nth root of a ; and a**, the 

nth root of the mth power of a (§871). 

333. Evolution is the process of finding the required root 
of a number. 

Evolution is the inverse of involution. Involution finds the product 
of equal factors, and evolution finds one of the equal factors of a product 

Monomials. 

834. Since (3 a^&s)^ = 3« a^^^ft^^^' (§ 319) = 27 aW, 
■</27a%^ = -v^ X a«-36^3 ^ 3 ^2^3^ 

Hence any root of a positive monomial is found by taJdrig the 
required root of its numerical coefficient, and then dividing the 
exponent of each literal factor by the index of the required root 

Write the indicated root of 

1. ViSbcM. 2. -y/W^^. 3. -y/U^HM^. 

335. Since any even power of ± a is positive (§ 316), any 
even root of a positive number is -}- or — . Thus, 

^1/I6^ = ±2a;. 

Since (+ a)2 = -f a^ and (— a)* = + a\ the square root of — a* is 
neither + a nor — o. The square root of — a^ can only be indicated 
(V— a^), and the same is true of any even root of a negative number. 
Such an indicated root of a negative number is called unreal or imagi- 
nary (§ 413). 

A number has as many roots as there are ones in the index of the root. 
Thus, v^lOas* = ± 2 a;2 and i V— 4 x*. The first two roots are real, and 
the second two roots (± V— 4 a;* = ± 2 ac^V— 1) are unreal. Only real 
roots are given in this chapter. 



§339.] INVOLUTION AND EVOLUTION. 195 

336. Since any odd power of + a is positive, and any odd 
power of — a negative, any odd root of a positive number is 
positive, and any odd root of a negative number is negative. 

337. It is thus seen that the sign of the even root of a posi- 
tive number is -\- or —, and that the odd root of any number has 
the same sign as the number itself, 

338. Since/^?Y=5^, ^1^ = ^=.^. Hence the root of 

\b) 6» \6» -y^ b 

an algebraic fraction is found by taking the required root of both 
its terms, and prefixing the proper sign. 

Write the indicated roots of 
4. V25a26^. 13. ->^-32aiV. 

6. V81 a'b'<^. 14. V^[^. ^^ ^ ^"y" 

6. ^256^. 15. VS^^^». 21. .^7^1^^ 

7. vu^w^, 16. v^;^^«. 



„^ g/ 32a^6^<> 



256m^^« 

8. ^-27mV. „/-,-^ * 22. a/^^- 

\ 17. -VS^V*. \ 2/^«" 

, -o 16 a^62 «&V* 

10. -v/-125mWs^. ^^- \o^^xl* 2^' \-^^' 



25a^ ^ sT 



11. </81^yV^' 3/ 216 a«a^ ^4 ^H^ 

12. -y/ - 243 a'^c^. ' \ 125 6^2* ' \ a«6*** 

Square Eoots of Polynomials. 

339. Since the square of a ± 6 is a* ± 2a6 + 6*, it follows 
that any trinomial is a perfect square, if, when properly 
arranged, its extreme terms are perfect squares, and its middle 
term twice the product of the square roots of its extreme 
terms (§ 186). Hence 

The square root of a trinomial that is a perfect square may 
be readily found by arranging the terms according to some letter, 
and then connecting the square roots of the extreme terms with the 
sign of the middle term. 



196 ALGEBRA. [§ 340. 

Find by inspection the square root of 

1. a?* — 4icy + 4y*. 5. 4a* — 4a + 1. 

2. 4aJ* + 4aj*2/ + 2^. 6. l-\-ey^ + 9f. 

3. a^-6a^f + 9j/*. 7. Oa^ - 2a?y + i2^. 

4. 9a'-12ay2 + 4y*. 8. i^^-a^ + y*. 

340. The second term of the square root of a trinomial 
which is a perfect square may also be found by dividing the 
middle term of the trinomial, properly arranged, by twice the 
square root of its first term. 

341. Since the square of any polynomial is composed of (1) 
the square of each of its terms, and (2) twice the product of 
each term multiplied in succession by the terms which follow 
it (§ 329), the square root of a polynomial that is a perfect 
square may be found by inspection, provided the given poly- 
nomial contains only two different powers of some particular 
letter. Thus, the square root of a* -|- 2 at + 2 ac + ^^ + 2 6c + c* 
is a + 6 -f c; and this may be verified by squaring a-\-h + c. 

342. It may assist, in finding the required root, if the poly- 
nomial be first arranged according to the descending powers of 
some particular letter. Thus, a* + 6* + c* -f 2 6c -f 2 ac + 2 aft, 
when arranged according to the descending powers of a, 
becomes 

a* -\-2a{h^ c)-h V + 26(c)+ <?. 

Find by inspection the square root of 

9. «* + y*-f-4-|-2iry — 4a; — 4y. 

10. a^-\-4:Qhj^-\'^a^z-\-^7?-\-4.y^^l2fz-\-l^y^ + 97? + 
2Az-\- 16. 

343. Since (a* + 2 a6 + 3 by= (a^ + 2aX2ab -\-Sb^-{-4t a^h^ 
+ 2(2a6 X 3 62) + 96*= a* + 4a«6 + lOa^t* + 12a6» + 96*, the 
square root of a* + 4 a^6 + 10 a*62 + 12 a6^ -h 9 6* is found by 
reversing the foregoing process, as is shown below. 



§343.] INVOLUTION AND EVOLUTION: 197 

11. What is the square root of 

a* -f 4a«6 + lOaV -f- 12ab^ + 96*? 

aH4a«6+10a262+12a68+96* | oH2fl6+36g , sq. root, 
a* 

'20^4-1 +4a«6+10a262 (1) 

(2a2+2a6)2a6== +4af6-f_4flW ^ 

2a2+| 6a2ft2_j.12a6S4.954 (2) 

(2aH4q5+362)3&2 = 6a26H12a6«-l-96* 

It is seen that the first term of the root (a^ is the square 
root of the first term of the polynomial ; that the second term 
of the root {-\-2ab) is obtained by dividing the first term of 
the first remainder (1) by twice the first term of the root, and 
the third term (3 b^ by dividing the first term of the second 
remainder (2) by twice the first term of the root. 

Note. If preferred, twice the first term of the root (2a^) may be 
considered the first " trial divisor ;" and twice the first two terms of the 
root (2a2 + 4a6), the second ** trial divisor." 

The first subtrahend (a*) is the square of the first term of 
the root ; the second subtrahend is twice the first term, plus the 
second term, multiplied by the second term [(2 a* -J- 2 ab) 2 db'] ; 
and the third subtrahend is twice the first two terms of 
the root, plus the third term, multiplied by the third term 
[(2a2 + 4a6 + 36^362]. 

For verification, square the root found by the binomial 
formula, or, if preferred, by § 329, rearranging the terms of 
the result, if necessary, and reducing to the simplest form. 

Find the square root of 

12. aj* — 4a^y + 6ic^2/^ — 4ir2/'4-y*. 

13. 9a^-12a^y -^34x^^-20x1^ + 251/*. 

14. 50 a + 15a^ + 25 -f a*- lOa^. 

15. l-10a: + 27aj2_10a^4-aj*. 

16. 4ar* + aV4-i2/'-4ar'3^ + 2aJ2/-aJ2^. 



198 ALGEBRA. [§ 344. 

17. l~4aj4-6y + 4aj*-12ajy+92^. 

18. 81 - 18a + a' + 186 -2a6 + 61 

19. ^a^ + ^a^ + ^a^^^f + y^^. 

20. ^-2xy + ^. 21. a^ + 3^+4«2a^ + ^-^- 

22. l + 2x + 2y-{-2z + it^ + 2xy + 2xz-^f + 2yZ'^i?. 

23. l-2a;-f 3a^-4aj»4-5a?*-4iB»4-3a^-2aj=^ + iB«. 

24. a«-6a*+15a«-20 + ^-4 + 4 

a* . a* a" 

344. To extract the square root of a polynomial, 

Arrange the terms according to the powers of some letter. 

Find the square root of the first term, and write it as the first 
term of the required root, and subtract its square from the first 
term of the polynomial 

Divide the first of the remaining terms by twice the first term 
of the root, and write the quotient as the second term of the root 

Multiply twice the first term of the root plus the second term 
by the second term, and subtract the product from the remaining 
terms; and so on until no term remains. 



Square Roots op Numbers expressed by Figures. 

345. The smallest integer expressed by one figure is 1, and 
the greatest is 9 ; the smallest integer expressed by two figures 
is 10, and the greatest is 99 ; and so on. 

346. The squares of the smallest and the greatest integers 
expressed by one, two, and three figures, are as follows : 

1*= 1 102= 100 1002= iQooo 

9« = 81 W = 9801 9992 = 998001 

It is thus seen that the square of an integer contains twice 
as many 07'ders as the integer, or twice as tnany orders less one. 



§ 350.] INVOLUTION AND EVOLUTION. 199 

347. The squares of the smallest and the greatest numbers 
composed wholly of units, or tens, or hundreds, are as follows : 

12= 1 10*= 100 100*= 10000 

9* = 81 90* = 8100 900* = 810000 

It is thus seen that the square of units is units, or units and 
tens; the square of tens is hundreds, or hundreds and thou- 
sands; the square of hundreds is ten-thousands, or ten-thou- 
sands and hundred-thousands ; and so on. 

348. It follows, that if an integer be separated into periods 
of two orders each, beginning at the right, there will be as 
many orders in the root as there are periods in the integer; 
and hence the square root of the lefi-Jiand period of the integer 
is the left-hand order of its square root 

1. How many orders in the square root of 64? Of 625? 
Of 1444 ? Of 273529 ? Of 45796 ? 

2. What orders in 273526 contain the square of the units 
of its square root ? The square of the tens ? The square 
of the hundreds ? What is the first or left-hand figure of the 
root ? 

3. How many orders in the square root of 145796 ? What 
is the first figure of the root ? 

349. An integer may be separated into periods of two 
figures each by beginning at units, and placing a dot over 
each alternate figure, thus, 145796; or by placing the dot 
between the periods, thus, 14*57*96. 

Separate into periods and give the first figure of the square 
root of 

4. 626. 6. 94249. 8. 5306845. 

6. 3026. 7. 492804. 9. 54756090. 

350. The square of an integer may be found by the binomial 
formula, as shown below. 



802 = 6400 

2x80x5= 800 

62= 25 



200 ALGEBRA. [§ 351. 

10. What is the square of 85 ? 

85 = 80 + 5. 
Let a = 80, and 6 = 5. 

Since (a + 6)2 = a^ + 2 a6 + h\ 

(80 + 5)2 = 802 + 2 X 80 X 5 + 52 = 7225 

It is thus seen that the square of any number composed 
of tens and units equals (1) the square of the tefiis, (2) plus 
twice the product of the tens multiplied by the units, (3) plu^ the 
square of the units. 

351. Since the square of tens gives no order lower than 
hundreds, and the product of the tens by units gives no order 
lower than tens, the tens and units of the root may be found 
as shown below. 

11. What is the square root of 7226 ? 

. . tu 

7225185 

a2 = 82 = 64_ 

2a = 2x80 = 160)825 

(2a + 6) 6 =(160 + 5) x 5 = 825 

The same result may be obtained by omitting the unit figure in l>oth 
the trial divisor (160) and the remainder (825): thus, 

. . tu 

7225186 

82= 64 
Trial divisor, 16)82|5 
165 X 5 = 826 

12. What is the square root of 104976 ? 

. . . htu 
f^ 10 49 76 1324 

(a = 3) 32 = 9 



Trial divisor, 2x3 = 6)14|9 
^(20x3 + 2)2 = 62 x2= 12 4 
(a = 32) Trial divisor, 32 x 2 = 64)2 57|6 
(320 X 2 + 4) 4 = 644 X 4 = 2 57 6 



§354.] INVOLUTION AND EVOLUTION. 201 

It must be observed that a represents first 3 hundreds, considered as 
3 tens with respect to the next figure of the root ; and that, in finding the 
third figure of the root, a represents 32 tens, the part of the root already 
found. 

352. Since .4* = .16, .04* = .0016, etc., the square of a deci- 
mal has twice as many decimal orders as the decimal; and 
hence the square root of a decimal has one half as many 
decimal orders as the decimal. A decimal is separated into 
periods by beginning at the decimal point, and pointing off to 
the right ; thus, .0625. A decimal cipher must be added if the 
decimal contains an odd number of orders ; thus, .6260. 

13. What is the square root of 13.3225 ? 

13.3225 [3.65 

32= _Q 

3x2= 6)4.32 
6.6 X 6 = 3.96 



3.6 X 2 = 7.2). 3625 
7.25 X .05 = .3625 

When an integer or a decimal is not a perfect square, its root may 
be found approximately by adding periods of decimal ciphers. Thus, 



V32 = V32.0000 = 5.65+. 

353. The square root of a common fraction is found by 
extracting the square root of each of its terms. Thus, VJ = f . 

When the denominator of a common fraction is not a perfect 
square, its square root can be found approximately by multi- 
plying both of its terms by the denominator, and then extract- 
ing the square root of both terms of the resulting fraction, 
carrying the root of the numerator to two or more decimal 

places. Thus,^=V^=-^=^=.75+. 

A common fraction may also be changed to a decimal, and the square 
root of the decimal found. 

354. When the right-hand period of a decimal contains 
only one order, a decimal cipher should be annexed, 4.322 
thus becoming 4.3220, 



202 ALGEBRA. [§ 355. 

Find, the square root of 

14. 69169. 21. 176.89. 28. f||. 

15. 94249. 22. 45.1584. 29. 272^. 

16. 57600. 23. .008836. 30. 1040^^. 

17. 210681. 24. .000625. 31. ^. 

18. 492804. 25. 75.364. 32. ^. 

19. 522729. 26. 586.7. 33. m|. 

20. 390625. 27. .056644. 34. ^V^- 

Find to three decimal places the square root of 

35. 2. 38. 3.5. 41. f. 44. f 

36. 3. 39. Q,^, 42. f. 45. ^. 

37. 5. 40. 0.9. 43. f 46. ^. 

Cube Koots op Polynomials. 

355. The process of extracting the cube root of a polyno- 
mial is readily derived from the formula 

(a ± 6)8= a8 ± 3a% ^^aV ± h\ 

This identity shows that a ± 6 is the cube root of 

It is observed that Va^ or a, is the first term of the cube 
root ; and ± 3 a^6 -^ 3 a^, or b, the second term of the root, 
3a^ being the trial divisor; and since 

±^a?h + 3a62 ± h^ = {Zd? + Sdb+hy), 

3 a^ + 3 a6 + 6^ is a complete divisor of the last three terms of 
the polynomial, h being the other factor. 

356. It is thus seen that the cube root of c?-\-Z a*6+3 a6*-f-6' 
may be found by taking the cube root of a^ for the first term 
of the root, and dividing 3 d?h by 3 a? for the second term of 
the root. Thus, 



§357.] INVOLUTION AND EVOLUTION. 203 

(af = aP_ 

Trial divisor, So" Sa^ft+Saft^+fts 

Complete divisor, 3a^+3a6-f ^^ 



(3d'+Sa^+b^)b=S^b±Sa^±^ 

1. What is the cube root oi a^ + 9 a^y -{- 27 ith/^ + 27 f? 

«6+9iK*yH-27a2y2+27 y8[a;f+3^ 

a* (a;2)8= ^6 

3a2(T. D.) ... 3aj* 

3a2+3a&+62(C.I>.) Sx^-{-9x^-{-9y^ 



9x*y+27xV+27y« 



(3a^ + 9g2y49y233y:^ 9a^y4.27g2y2.|_27y8 

357. A cube root having any number of terms may be found 
in like manner if it be observed that each successive trial 
divisor is three times the square of the part of the root 
already found ; and that each successive divisor is completed 
by adding to the trial divisor (1) three times the product of 
the root term last found multiplied by the part of the root 
before found, and (2) the square of the term kist found. 

2. What is the cube root of 

8a^- 36ic* + 66 a?* - 63a^ + 33a^ - 9aj + 1? 

I2a;2-3a;+l 



8ir«-36iB6+66aj*-63x8+33x2_9a;_|_i 
(2a;2)8= 8^ 



(T. D.) 3(2a;2)2= 12a^ | -36a*+66rK*-63a^ 



(12«*-18a;8+9«2)(_3«)= -36a^+64a^-27a^ 
(T.D.)3(2x2-3x)2= 12a^-3 6a;84-27a;2 | 12a:*-36«8+33a;2-9a;+l 
(12x4-36«8+27xH6a;2-9x+l)l= 12a:*-36a^4-33g2-9a;+i 

Find the cube root of 

3. iB« + 6aa^ + 12aV+8a«. 

4. 8a^-12aar* + 6a2aj-al 

6. 125 a« - 225 a%^ + 135 a%* - 27 b\ 

6. a«4-3a* + 6a* + 7a3 + 6a2 + 3a + l. 

7. aj« + 6a^-40iB8 + 96aj-64. 



204 ALGEBRA. [§ 358. 

8. 8aj«-12aj«-54aJ* + 59aj«-f 135ar^-75aj-125. 

9. l+9» + 18a^-27aj3-54aJ*-f 81a?5-27i««. 

10. aj8-6aj + — --,• 

X or 

11. a8-3a* + 9a-13 + — -^ + -- 

a (T a^ 

12. 27 a«-108 a'b+90 a'b^+SO a^ft^-BO a26*-48 a6*~8 b^ 

13. 8 a:^ - 36 ar'^/ + 66 xY - 63 xY + 3Sa^y^ -9xf + f. 

14. 27 a^ - 54 aar' + 63 aV - 44 aW + 21 aV - 6 a^a? + a^. 

358. To extract the cube root of a polynomial, 

Arrange the terms according to the powers of some letter. 

Find the cube root of the first term, write it as the first term 
of the root J and subtract its cube from the polynomial 

Divide the first of the remaining terms by three times the sqvxire 
of the first term of the root (trial divisor), and write the qriotient 
as the second term of the root. 

To the trial divisor add three times the product of the fifst 
term multiplied by the second, and the square of the second; and 
then multiply the complete divisor thus formed by the second term 
of the root, and subtract the product from the remaining terms 
of the polynomial. 

Proceed in like manner until all the term^ of the polynomial 
are used. 

Cube Eoots of Numbers expressed by Figures. 

359. The cubes of the smallest and the greatest integers 
expressed by one, two, and three figures, are as follows : 

13= 1 103= 1000 1003== 1000000 

9^ = 729 99^^ = 970299 999^ = 997002999 

It is thus seen that the cube of an integer expressed by one 
figure contains from one to three orders; that the cube of an 
integer expressed by two figures contains from four to six 
orders; and that the cube of an integer expressed by three 
figures contains from seven to nine orders. 



§364.] INVOLUTION AND EVOLUTION. 205 

360. It follows, that the cube of an integer contain^ three 
times as many orders as the integer, or three times as many 
orders less one or less two, 

361. The cubes of the smallest and the greatest numbers 
composed wholly of units, or tens, or hundreds, are as follows : 

13= 1 103= 1000 1003= 1000000 

^ = 729 903 ^ 729000 900^ = 729000000 

It is thus seen, that if a number be separated into periods 
of three orders each, beginning at the right, the first period 
will contain the cube of the units of its cube root ; the second 
period, the cube of the tens of its cube root ; the third period, 
the cube of the hundreds of its cube root ; and so on. 

362. It follows, that the cube root of an integer contains as 
many orders as there are periods in the integer, and the cube 
root of the left-hand period of an integer is the left-hand term of 
its cube root, 

363. An integer may be separated into periods of three 
figures each by placing a dot over the first or units, fourth, 
seventh, etc., orders, thus, 48228544; or by placing a dot 
between the periods, thus, 48*228*544. 

Separate into periods and give the first figure of the cube 
root of 

1. 42875. 3. 117649. 5. 274625. 

2. 91125. 4. 185193. 6. 9405424. 

364. The cube of a number may be found by the binomial 
formula (a ±lif = a^ ±3a% -\-^ab^ ±W, d^ shown below. 

7. What is the cube of 85 ? 

808 = 512000 

85 = 80 + 6. 
Let a = 80, and & = 5. 

Since (a + &)« = a^ + 3 a^ft -\-Sab^+ h^ 



3 X 802 X 5 = 96000 

3 X 80 X 52 = 6000 

58 = 125 



(80 + 6)8 = 808 + 3 X 802 X 5 -f 3 X 80 X 52 + 68 = 614125 



206 



ALGEBRA. 



[§ 365. 



365. It is thus seen that the cube of any number composed 
of tens and units is equal to (1) the cube of the tens, (2) plus 
three times the product of the square of the tens multiplied by 
the units, (3) plus three times the product of the tens multiplied 
by the square of the units, (4) plus the cube of the units. 



Find thus the cube of 

8. 82. 10. 77. 

9. 68. 11.. 66. 



12. 104. 

13. 125. 



14. What is the cube root of 262144 ? 



«»+ (3a2 + .3a6 + &2)6 



a* 



(T. D.) 



3a2 



68 = 

. . . 3 X 602 = 10800 

3 a6 . 3 X 60 X 4 = 720 

62 . . 42= 16 



t u 
262144 |64 

216 



46144 



(3a2 + 3a6 + 62)5 



= 11536 X 4 = 46144 



It is seen that the first trial divisor is 3 x 6O2 = 10800, and that the 
complete divisor is 11536. 

15. What is the cube root of 16387064 ? 



a« 



(2)8 = 



16387064 [264 
8 



3a2 3 X 202 = 1200 8387 

Sab . . 3x20x5= 300 
62 62 = 25 

(3a2 4-3a6 + 62)6 

3a2 3 X 2602 

3a6 . . 3 x260 x4 

62 42 

(3a2 + 3a6 + 62)6 = 190516 x 4 = 762064 



1525 X 5 = 7625 
187500 
3000 
16 



762064 



It should be noted, that, in finding the first two terms of the root 
(25), a in the formula denotes hundreds, and 6 tens (hundreds being 
considered tens, and tens units); but, in finding the third term (4), 
a denotes 25 tens, and 6 the units sought. 



§ 867.] INVOLUTION^ AND EVOLUTION. 207 

The process may be continued to any number of terms in the root 
by observing, when finding a new term, that a in the formula denotes 
the part of the root already found. 



366. Since .!« = .001, .Ol^ = .000001, etc., the cube of a deci- 
mal has three times as many decimal orders as the decimal ; 
and hence a decimal is separated into periods of three orders 
each by beginning at the decimal point, and pointing off to the 
right, thus, .015625. . If the last period does not contain 
three figures, decimal ciphers must be added ; thus, .262500. 



367. Since the cube of a fraction is obtained by raising each 
term to the third power (§ 315), the cube root of a fraction is 
found by eoctracting the cube root of each of its terms. If the 
denominator of a fraction is not a perfect cube, its approxi- 
mate cube root may be most readily found by first changing 
the fraction to a decimal. 

Find the cube root of 

16. 42875. 23. 97.336. 30. 5fjf. 

17. 185193. 24. .097336. 31. 37^. 

18. 3048625. 25. 1953.125. 32. ^^ 

19. 48228544. 26. 67.419143. 

20. 34328125. 27. 28.094464. ^3. 



1.728 
1.5 



.216 
21. 27270901. 28. ^VA- « ^ 



22. 74.088. 29. 1444. .0625 



Find to three orders the cube root of 

35. i^. 38. -g-. 41. ■^^. 

36. y. 39. -g-. 4^. YTj". 

37. 10. 40. i. 43. ^. 



208 ALGEBRA. [§ 368. 

368. To extract the cube root of a number, 

Begin at units, and separate the number into periods of three 
figures each. 

Find the greatest cube in the left-hand period, and write its 
cube root as the first term of the required root. 

Subtract the cube of the first term of the root from the left- 
hand period, and to the difference annex the second period for 
a dividend. 

Divide this dividend by three times the square of the first term 
of the rooty tvith ttvo ciphers annexed {trial divisor), and write 
the quotient as the second term of the root. 

To the trial divisor add three times the product of the first term 
multiplied by the second, and the square of the second; and 
then multiply the complete divisor thus formed by the second term 
of the root, subtract the product from the dividend, and to the 
difference annex the next period OjS another dividend. 

Proceed in like manner until all the figures of the given numr 
ber are used. 

369. The fourth root of a number may be found by extract- 
ing the square root of its square root ; and the sixth root, by 
extracting the cube root of its square root, or the square root 
of the cube root. The cube root of the square root is prefer- 
able in practice. Thus, 

\/G25=VV625=V25 = 5; \/4096 = ^^ V4096 = "^ = 4. 

Find, as above, the fourth root of 

44. 16 a* + 96 a^b + 216 a'b^ + 216 ab^ -\- 81 b\ 

45. ic8-8a;^-f-16aJ«+16ar^-56a;*-32aj3+64a^ + 64aj + 16. 

Find the sixth root of 

46. a^ + 6a^2/-f 15iC*2/^ + 20a^2/8 + 15ajy-f ean/' + Z. 

47. 1 + 12a + 60a2 + 160a3 + 240a* + 192a^ 4- 64a«. 



§ 374.] RADICALS. 209 



CHAPTER XIII. 
RADICALS. 

370. A radical is the indicated root of a number ; as Voib. 

The term radical is also applied to expressions that contain a radical ; 
as 3 aVai), and 3 a + Vab (§§ 375, 394). 

371. If the indicated root can be exactly obtained, the 
radical is called rational; if the indicated root cannot be 
exactly obtained, the radical is called irrational or a surd. Thus, 
a/25 and Va^ are rational, and V5 and -v^a-j-ft are surds. 

Roots may also be indicated by fractional exponents, as 6*, (a + 6)* 
(§ 332) ; and roots thus indicated may be rational or irrational. Thus, 

6^ and {a + hy are surds. 

Irrational numbers are also called incommensurable^ since they have 
no common measure with unity. 

372. It is to be observed that algebraic surds may become 
rational when numerical values are assigned to the letters. Thus, 
if we make a = ^, and 6 = 3, the surd Va-h h becomes rational. 
Surds that can be rationalized are said to be surds in form, 

373. The degree of a radical is indicated by the index of the 
radical sign. Thus, VoS (index 2 understood) is a radical of the 
sec(ynd degree ; Va -f 6, a radical of the third degree ; and so on. 

Radicals of the second degree are also called quadratic 
radicals ; and those of the third degree, cubic radicals. 

The degree of a radical is also called its order. Since surds are a class 
of radicals, what is true of radicals generally is true of surds. 

374. The coefficient of a radical is the factor placed before 
the radical part. Thus, in the expression 6^ a — h, 5 is the 

white's alo. — 14 



210 ALGEBRA. [§ 375. 

coefficient of Va — 6; and in the expression 7a^/bx, 7 a is 
the coefficient of -Vbx, 

375. When a radical contains no coefficient (except 1 under- 
stood), it is said to be eivtire; and when it contains a coefficient, 
it is said to be mixed. Thus, Voi and Va — a? are entire radi- 
cals, and 2V5 and 5 a Va? — y are mixed radicals. 

The coefficient of a radical is called the rational factor ; and the radical 
part, if a surd, the irrational factor. 

376. Similar radicals are those which have the same index, 
and the same number under the radical sign ; i.e., have their 
radical parts identical. Thus, 2 V3 and aV3 are similar ; so, 
also, are 3^—2 and 2a^— 2. 

377. Since Va x \/h= Va6, and, conversely, Va6= Va x V&, 
it follows (1) that the product of the same roots of two factors 
eqvxds the same root of their prodiict; and conversely (2) that 
the root of any product equals the product of the same roots of 
its factors. 

378. This principle enables us to reduce a mixed radical to 
an entire radical, and an entire radical to a mixed radical ; also 
to make other important reductions of radicals in degree and 
in form, as shown below. 



REDUCTION OF RADICALS TO EQUIVALENT RADICALS OF 

DIFFERENT DEGREE. 

379. Since a = Va^, or Vo?, or Va"*, it follows that any 
rational number may be changed to an equivalent radical of 
any degree by raising it to the power corresponding to the index 
of the radical, and placing the result under the radical sign. 



Thus, 005 = ^aV; 3~\gT; and a — 6 = V(a — 



by 



§ 383.] 



RADICALS. 



211 



380. In like manner a radical of any degree may be changed 
to an equivalent radical of a higher degree by mvltiplying the 
index of the radical, and the exponent of each factor under the 
radical sign, by the same number. Thus, 

V5 = -^= -^5/125; ^/a^ = ^/a^', and Va + b = V(a + b)\ 

381. Conversely, any radical may be changed to an equiv- 
alent radical of a lower degree by dividing the index, and the 
exponent of each factor under the radical sign, by the same 
number. Thus, 

■y/a^=-s/^'^ </(a -h 6)- = VoT^ ; and -v^o^ = Vo^. 

Reduce the following radicals to equivalent radicals of the 
indicated degree : 

1. a6 to third. 

2. oj — 2^ to second. 



3. ^ to fifth. 
or 

4' Va» to sixth. 

5' ->/x-\-y to sixth. 



6« V(a — xy to nth. 
7« VaS to mth. 
8- -y/c^b^ to second. 



^' ^(^ — yf to nth. 

0. V^,..... 



{^ + yy 



to second. 



1. - to fourth. 
x 



If a; — 



to sixth. 



x — y 



3. "\/a*V*d^ to nth. 






6) 



3 



to second. 



382. Radicals of different degrees may be reduced to 
equivalent radicals of any degree which is a multiple of their 
indices. Thus, VB and -y/l may each be changed to the sixth 
degree, as above; and -y/ac, -Va — b, and Va 4- b may each 
be changed to the twelfth degree. 

383. This process is called the reduction of radicals to a 
common index. When the common index is the L. C. M. of the 
indices, it is a common index of the lowest degree. 



212 ALGEBRA. [§ 384 

Reduce to a common index of the lowest degree 
16. V6, n, and ^10. ^^ & J2 ^^ ^ 

16. Var^, -Vxy, and wa^j^. r r~ r~ 

17. ^, .5/26, and ^^«. ^^^ \a' \h "^^ \? 

20. a?, -y/och/f and VaJ^^. 

21. -y/x — y and ^/x + y, 

22. a — 6, -^(a — 6)*, and -v/(a — &)^. 

23. V^ft, lj^, ^I^. 



+ 6y 



REDUCTION OF RADICALS TO SIMPLEST FORM. 

384. A radical is in its simplest form when its radical part 
is integral, and contains no factor which is a perfect power of 
the same degree as the radical. Thus, Va^ — 6* and 3Va^ 
are radicals in their simplest form. 

385. When the number under the radical sign contains a 
factor which is a perfect power of the same degree as the 
radical, the radical may be reduced to its simplest form by 
removing such factor from under the radical sign, and making its 
proper root the coefficient, or a factor of the coefficient. Thus, 

■\/25 a^x = V25 a^ x x = -\/25 a^ x Va = 5av^; 
3^/1^ = 3^/^ X \/P = 3a\/P. 

Reduce to simplest form 

1. V75. 4. fv^l92. 

2. -v'320. 6. V32^*. 

3. 2</80. 6. 3a^6256y. 



§387.] RADICALS. 213 



7. V125 a» - 50 a*6. 10. V(x- y)(x''-f), 

8. ^16a5y-24aY- H- V^C^ + y)'(«'-yO. 

9. V(a4-&)(a^-2>^- 12. (« + y)VaJ^- 2ic*y + ary^ 

13. |(a 4- 5) V3a2&2 - 3006^ 4- 75 6*. 

14. 5 a Va? — 2 ajy + o^. 

15. (a + y)Va* — icV- 

386. When a radical is fractional, it may be reduced to its 
simplest form by multiplying both terms of the fraction by such a 
number as wiU make the denominator a perfect power of the 
degree of the radical, and then proceeding as above. Thus, 

Reduce to simplest form 

16. V^. 18. ^. 20. 10 Vf a^x, 

17. ^. 19. 3^. 21. lOa^fe^— ?-. 

^ 25 aW 




23. -i-J ^-^'. 26. !c'-«'\/ ^ 

24. 6aJ^ni. 27. £z:lj/27(^ + l). 

\3(o-6) 3 >( (a;-l)» 

387. When the quantity under the radical sign is a perfect 
power whose exponent is a factor of the index, a radical may 
be simplified by dividing both the index and the exponent by the 
common factor. Thus, 

■V^='^/c^; ■^(a + W = Va + b; and ■^(a-by=(a-by. 



214 ALGEBRA. [§ 388. 

Eeduce to simplest form 
28. </IOO. 32. -^(a + &)'(a - 6)«. 






29. V125aV. 33. V4 a* - 24 a6 + 36 &*. 

30. </(a' - by. 34. '^a^y^\x-yy. 



31. A/(a2 - 6*)Xa + i>)*. 35. Va^ft^a + 6)^. 

INTRODUCTION OF COEFFICIENTS UNDER THE RADICAL 

SIGN. 

388. The coefficient of a radical may be placed under the 
radical sign by raising it to the power corresponding to the index, 
and introducing the residt under the sign as a factor. Thus, 

6 Vai = V25 X VoS = -^250^] 

5 a -y/xy = ■y/25c? x y/xy = V25 a^xy. 

It will be observed that this is the converse of § 385. 
Thus, V4 a^oj = 2 a Vx, and, conversely, 2 a -y/x = V4 a*a?. 

Introduce under the radical sign the coefficient of 

1. 2 v^oS. g m + 71 / m — n 

2. -3aVi06. m-n\m + n 

3. |V5^. 9. _J_^w3_^8. 



4. -V^. 
2 



m — w 



^4aV 



4a^ar^ 
6. 3ic ' 



13. («-&)V-^- — 
^a — 6 a — 



b 
6 



§ 390.] RADICALS. 215 

ADDITION AND SUBTRACTION OF RADICALS. 

389. Since the coefficient of a radical denotes the number 
of times it is taken, it is evident that the sum or difference of 
two similar radicals is found by prefixing the sum or difference 
of their coefficients to the common radical part. Thus, 

7Va6 + 5Va& = 12 Va6; and 7 Va— 6 — 5Va — 6 = 2^ a — 6. 

390. If the radicals to be added or subtracted are dissimilar, 
they must first be changed to similar radicals (§ 376). When 
this cannot be done, their sum or difference can only be indi- 
cated by the proper sign. 

1. Add 7V8, 3V32, and - SVJ- 

7V8 = 7Vr>r2= 14 V2 
3 V32 = 3Vi6x^ = 12V5 

- 8Vi = - SVfx^ = -4 V2 

Sum= 22v^ 

2. Simplify V8(a - h) -f- 4 Vl8(a - h) - 12Vi(a -b). 

v'8 (a - 6) = V4 X 2 (a - 6) = 2V2(a-6) 

4 Vl8 (a - 6) = 4 V9 x *2 (a - 6) = 12V2(a-6) 

- 12Vi(a-6) = - 12V32j(a-6) = - 3\/2 (a - 6) 

Value = llV2(a-&) 

If the given radicals are of different degrees, they most first be reduced 
to the same degree (§§ 380, 381). 

Simplify 

3. Vi8 + V32. 8. 2</250 + 3</54. 

4. V54 + 2V294.> 9. -y^logo ~.>/iO +-^135. 

5. 12V20-3V45. 10. 3 V80 + 6 V45 - 2 Vi25. 

6. 4V|-6Vp. 11. VJ-2Vi-VS. 

7. 16V3^-4V|-Vl|. 12. V2+V8-VJ-Vi. 



216 ALGEBRA. [§ 391. 

13. 4Vi + iVi-3V^. 20. |Vl8-VJ+Vi+3VS. 

14. -V\+V^+-^^. 21. 3ay/^-5aVxy*- 

15. </48+</^ + 8</^ 22. 5a'V^-5axV^. 

16. -^270-5^80+^640. ^^ 3a»<^-3a^^^. 

^a >a^ ^ar 

25. 3^c - 5-v/? + 2^. 

26. 5 V(a — xf — 2 axVa — a?. 



- 4-4-4 



391. To find the sum or difference of radicals, 

Reduce the radiccUs, if necessary, to similar radicals, and 
then prefix the sum or difference of the coefficients to the radical 
part. 

If the radicals cannot he made similar, indicale their sum 
or difference by the appropriate sign, 

MULTIPLICATION OF RADICALS. 

392. Since a^/a x b^=a xb x -\/a x V& = ab^s/ab, it fol- 
lows that the product of two or more radicals of the same 
degree equals the product of their coefficients prefixed to the 
product of their radical factors. 

If the radicals are of different degrees, they must first be 
reduced to equivalent radicals of the same degree. 

1. What is the product of 6V8 and 7V6? 



6V8 x7\/6 = 6x7x VSx^ = 42\/l6~x3 = 168V3. 
2. What is the product of 3 V2 and 5^4 ? 

.-. 3V2 X6\^ = 3\^8x6v^i6 = 3x6\/8^ri6 = 16v^6rx2 = 30v^. 



§394.] RADICALS. il7 

3. What is the product of Va and -y/h? 

Simplify 

4. 6V8x7V2. 18. 3Vaax2V5^. 

5. 5vT0x2vl5. 19. 3V^ X 2V5a» X 3Va. 

6. ^^v/12X|-^. 



- ^g X ^>§- 



8. 2V|x3Vf ,^. 2aM^3JIg-. 

9. V|xA/fxV|. ^52 >i8a!V 

10. 4-v^ X 5\/9 X 2\/4. 22. 3aV^x2bVa\ 



11. 12\^ X 2<^ X 3^^. 

8^ 



23. 3\P^ X 4-v/a - «. 
^a — oj 



12. VlBxViO. 

13. 2 V3 X 3v^ X ^. 24. aVS x 6^? x c^. 



14. 2^3x2V4x3V2. 25. V3^x ^5/4^x^2^^. 

16. 2vi X 3v^ X V|. ^aj2 — 2^ 

17. 4Vo^x3</^. 27. a/5 X \/^^^25 X ^/iO. 



393. To find the product of two or more radicals, 

Reduce the given radicals, if necessary, to equivalent radicals 
of the same degree; then prefix the product of the coefficients to 
the product of the radical factors, and reduce the resulting radical 
to its simplest form. 

394. A compound radical is a polynomial that has one or 
more radical terms; as, 2 Va + 3V6, a — V3^, Va — Va — &, 
and -\Ja-\- 6— Va— 6- 



218 ALGEBRA. [§ 395. 

28. Multiply x + 2V3 by a; -VS. 

X +2V3 
« - V3 



x2 + 2 X VS 

- gV3-2VP 

The product of two binomial radicals may be written directly as 
in § 166. 



Multiply 

29. V5 — 3 by Va 4- 2. 

30. 2 Va - 4 V3 by 3 V« + 2 V3. 

31. 2V5 + 3V2 by 3V5 - 4V2. 

32. 2Va-3V6 by VaH-A/6. 

33. 4Va+V6 by 4Va + V6. 

34. a: + V^ + 2/ by V« — V^. 

35. aH-aV3H-l by aV3 — 1. 

36. -\/3 + 2-\/2 by 2-\/3 + ^. 

37. 2V| + 3Vi + l by 3Vi-2VJ. 

395. When two binomials involving radicals differ only in 
the sign of a radical term, they are said to be conjugate ; and 
their sum and product are rational. Thus, 

(a + V6)(a-V6)=a2-6, and (Va-A/&)(Va+V6)=a-6. 

Multiply 

38. 2^—^ by 2Va + V6. 

39. -yfx + 2 V^ by V5 — 2V^. 

40. 3 V5 + 2 V3 by 3V5 - 2V3. 

41. ^v^-Vl2 by ^VS+Vi2. 

42. Va;-V21 by V^TV^I. 

43. l+Vaj + l by l-Va + l. 

44. Va+^— Vo^^ by Va 4- a; 4- Va — ox 



397.] RADICALa 219 



DIVISION OF RADICALS. 

396. Since ay/x x h-Vy = db^^/xy, db -y/xy -j- hVy = a-Vx ; 
and hence the quotient of two radicals of the same degree 
equals the quotient of their coefficients prefixed to the quotient 
of their radical factors. 

If the radicals are of different degrees, they must be reduced 
to radicals of the same degree before dividing. 

1. Divide 12V75 by 5V3. 

12 V76 ^ 6\/3 = ^VJ^ = V^^ = 12. 

2. Divide V^ by >^. 

Divide 

8. 6Vl08 by 3V6. 12. V3f by ^. 

4. 3V8 by 6V27. 13. -\/2^ by V2}. 

6. 3V6 by V8. 14. 2V5 by ^5. 

6. 3vl^ by 2V|. 15. 15V^ by ^Va^. 

7. 12 Vf by 3V|J. 16. 2aV^ by 5aj^v^^. 

8. ^A^ by ^^. 17. Sa^v^'^ by 10 xV^. 



9. 6V8-6V2 by 3V2. is. 5V? by 4\/?. 

10. Vl by ^6. 19. 3aVa«5 by 2a-\/cM. 

11. ^^I2 by Vs. 20. Va^-y* by V«^. 

397. To find the quotient of two radicals, 

Reduce the given radicals, if necessary, to equivalent radicals 
of the same degree; then prefix the quotient of the coefficients 
to the quotient of the raduxU factors^ and reduce the remiUing 
radical to its simplest form. 



220 ALGEBRA. [§ 398. 



BATIONAlilZINO THE DlYISOB. 

898. If the denominator of a fraction is a radical, the denom- 
inator may be made rational by multiplying both terms of the 
fraction by such a number as wiU make the denominator a per- 
fect power of the degree of the radical. 

Thus, X=AxVL=?V3^^. 
V3 V3xV3 V9 

2 X Va^ '^^^^ '^^^^ 



a' :5 V a 



^/a ^/ax^/c^ ^/^ ^ 
Reduce to equivalent fractions with rational denominators 

1. A,- 6. ^-^. 9. ' 



V3 V6 </S^' 

2. 2/1. « VlO 10. ^ 



V3  2Vfi 2V6 



u> 


V6 


6. 


Vio 

2V6 


7. 


3 


8. 


5 



3. i5.. ^ 3 11. " 



V2 " m </f 



4. ^. 8. -^. 12 

Vl2 V3s^ 



■4 



399. In like manner the quotient of two radicals may be 
found by rationalizing the divisor, and then reducing the quo- 
tient to its simplest form. Thus, 

400. When a divisor is a binomial of the form Va ± V&, it 
may be rationalized by multiplying both dividend and divisor 
by the conjugate of the binomial divisor (§ 395). Thus, 

V3 ^ V3x(V3+V2) ^ 3+V6 ^3^^/g 
V3-V2 (V3-V2)(V3-f-V2) 3-2 



§ 401.] RADICALS. 221 

Eationalize the divisor and simplify 

13. V8l^V3. 23 V8 



14. Vios^Ve. V5-I-V3 

3+V5 



15. V48-f-Vi2. 24. 

16. 3V8-5-V6. 

17. Vi5-^2V5. ^^• 



3-V5 

V3+V2 



V3-V2 

18. (V3-V2)--V6. 2^ V^+V6 

19. V3^Vl0. Va-V6 

6 27. ^V^ + ^V^ - 

' 3V3-2V2' 

V7 
5ViO + 2V3 
2V3-f 3V2 
3V3 4-4V2 



A*vr. 


6-V7 


21 


1 


ax. 


Vio-3 


29 


21 


40^. 


Vio+V3 


L _ 


1 



28. 



29. 



30. 31. -' 32. . 

I-V24-V3 1+V2-V3 2+V3+V5 

Suggestion. In Example 30, multiply both terms by (1 — V2) — V3, 
and then multiply both terms of resulting fraction (simplified) by — 2\/2. 

401. The above method of rationalizing the divisor has 
special value in finding the approximate numerical value of a 
quotient when the divisor is a surd^ as shown below. 

33. Find the approximate value of 4 -«- VS. 

J_^4V3^V38==?:M± = 2.309+. 
V3 3 3 3 

5 



34. Find the approximate value of 



4-V2 



6 ^ 6C4 + \/2) - 2Q + ^^ - 2Q + ^^^ - 1.0336 1, 
4-V2 (4->/2)(4 + >/2) 1^-2 14 



222 ALGEBRA. [§ 402. 

Find to three decimal places the value of 



36. 


2 
V8 


86. 


10 

V5 


37. 


6 
Vl2 


3ft. 


6 



<IQ 


3 


Oo» 


V2-2 


AO 


V3 


t\9* 


2-t-V3 


41 


2-V3 


tbx. 


3-V2 


iiO 


V5-V3 



d^ 


6 


%o. 


V3+V2 


44 


V6+V2 


w. 


V6 


dfi 


3+V5 


%o. 


3-V6* 


46. 


V8 



V27 V5 + V3 V8-V2 

INVOLUTION AND EVOLUTION OF RADICALS. 

402. Radicals may be raised to any power by substituting 
fractional exponents for the radical signs, and then proceeding 
as in the involution of numbers with positive integral expo- 
nents (§ 319). The roots of radicals may be found in like 
manner. For other operations with fractional exponents see 
Chapter XIV. 

1. Find the square of 4Va. 

(4v^)2 = (4 a*)2 = 16 a* = 16 v^. 



2. Find the cube of 3 Vod*. 

(3 VaP)8 = (3 aM)8 = 27 Jb^ = 27 Vo^S*. 

Raise to the indicated power 

3. {-Vsy. 7. (-v^)'. • 11. {-V^^ 

4. {5</3y. 8. (^*. 12. (2V^6^*. 

5. {</i2y. 9. (V^«. 13. (4^;/a"=^)*. 

6. (V32)\ 10. (2^/a^*. 14. (xVyy. 
15. Extract the cube root of SVo^. 

v^8Vax8 = (8 aW)^ = 2 a^x* = 2-^^. 



§405.] RADICALS. 228 

16. "v^VeJ. 19. V26\/6. 22. ^^^/^^. 

17. "V^VSl. 20. "V^Vo^. 23. ^a-Wa. 



18. Vl25V8. 21. Vo^Va?. 24. ^^o^. 



Square Roots op Binomial Surds. 

403. A binomial one or both of whose terms are surds is 
called a binomial surd. Thus, a -\- V6, Va — b, and Va ± V6 
are binomial surds. 

The finding of the square root of a binomial surd which is the sum ol 
a rational number and a quadratic surd, as a i: Vb^ is of sufficient utility 
to justify a brief treatment here. 

404. Since ( V5 ± V sy = 5 ± 2 VlS + 3 = 8 ± 2 VlS, then, 

conversely, Vs ±2Vi5=V6 ± V3; and hence 8 ± 2^/l5 is 
a perfect square. It is thus seen that some binomial surds 
of the form a ± Vb are perfect squares. 

405. A formula for finding the square root of such binomial 
surds may be obtained as follows : 

(Va-hVS ± Va - Viy = 2a± 2Vc?^^. 



.'. Va+V6+Va-V6=V2a + 2V^^, (1) 

and Va+V6-Va-V&=V2a-2V^^. (2) 

Adding (2) to (1), member to member, and dividing by 2, we 
have 



and subtracting (2) from (1), and dividing by 2, we have 



(3) 



V^3;^ Jg+V^'-ft Jg-Va'-ft. 

2 2 



(4) 



224 ALGEBRA. [§ 406. 

406. It is evident from Formulas (3) and (4) that the square 
root of a ± Vft may be readily found if a? — h is a perfect 
square. This may usually be determined by inspection, and 
the square root of the binomial is then found by substituting 
the values of a and Va^ — h in the proper formula. The 
method is not practicable when a^ — b is not a perfect square. 

1. Find the square root of 6 4- VlT by formula. 

Va2 - 6 = V62-11=V26 = 5, 

whence V^Tv!T=V^ + a'^ = V| + A| = i>^ + i^- 

2. Find the square root of 17 — 4 Vi5. 

^17-4Vl5 = Vl7-V^; V172 - 240 = VS = 7 ; 
whence Vi^ ^4^/l^=J^^^-^^-J^^^^:^ = Vl2^^/E==2y/S-^^ 

Find the square root of 

3. 64.V2O. 8. 7 + 2VlO. 13. 6a-\-2aV5. 

4. 10+Vi9. 9. 11-V72. 9 

14. 



5. 7 + 4V3. 10. t^+VJ. 6 + 2V6 

6, II-V2I. 11. f-|V7. „ 121 

ID. 



7. 21 + 4V5. 12. | + iV24. 9-Vi7 

407. The square root of a binomial surd may also be found 
by first so reducing the surd term that its coefficient is 2 ; then 
separating the rational term into two parts such that twice the 
product of their square roots equals the surd term ; and then 
connecting such square roots with the sign of the surd term (§ 339). 
Thus 



(1) V6+vTi = V-V- +2Vv-+^=V v: + V|=^V22 + ^V2. 

(2) Vl7-4Vl5 = Vl2-2V6()-f-5 = Vl2-V5 = 2V3-5. 
Find by this method the square root of 3 to 15 above. 



§409.] RADICALS. 226 

Another Method. 

408. If a ± -y/b = c± V5, and a and c are rational, and Vft 
and V^ are surds, then a = c, and b = d. 

For, if possible, let a = c -|- a? ; 

then, substituting, c + x± -y/b = c ± V5 ; 
whence x ± V6 = ± V5. 

Squaring, a^ ±2 x-y/b -h 5 = d ; 

transposing, o^-fft — (Z = :f2 oj V&, 

which is impossible, since a rational number cannot equal a 
surd; and hence 

a = c, and V6=Vd, or 6 = d. 

409. The square root of a binomial surd of the general form 
a ± VS may be found as follows : 

Suppose Va ± Vft = VS ± V^ ; (1) 

squaring, a ± -y/b = a? ± 2 V^ H- y. 

Hence, by § 408, we have 

a? + y = a, and 2 V^ = Vft ; 
or 3^ + y = a, and 4iBy= 5. (2) 

The value of v a ± Vft can now be found by finding the values 
of X and y m equations (2), and substituting the same in (1). 

16. Find the square root of 15 + 2 V56. 

Let Vl6 + 2y/m = Vx + Vy ; 
squaring 16 + 2V56 = « 4- 2Vicy -|- y ; 

whence « + j/ = 16, and 2Vay = 2\/56; 

or X 4- y = 16, and ay = 56. .•. a: = 7 ; y = 8. 

Hence Vl6 + 2V'66 = V? + VS. 

Find the square root of 

17. 7-h2VlO. 19. 11 + 2V30. 21. 16 + 2 VSK 

18. 6+V20. 20. 12-6V3. 22. 6-2-/^. 

whitk's alo. — 15. 



226 ALGEBRA. [§ 410. 

EQUATIONS INVOLVING RADICALS. 

410. An equation in which the unknown quantity occurs 
under the radical sign is called a radical equation. 

411. The more common methods of solving radical equa- 
tions are indicated below. 



1. Solve the equation V«* — 15 -f- a? = 5. 



Transposing jc, Vac* — 16 = 6 — ac ; 

»juaring, a? - 16 = 26- lOz + afi; 

transposing, 10 x = 40 ; 

whence * « = 4. 

2. Solve the equation VaJ + 7 = 7 — a^. 

Squaring, a; + 7=49 — 14>/5c + x; 

transposing, 14 v^ = 42 ; 

dividing by 14, Vx = 3 ; 

squaring, x = 9. 

3. Solve the equation V« — a= Va5*4-«aj. 

Raising Vx- a to 4th degree, v^(x — a)^ = -v^x* + ax ; 

raising to 4th power, x* — 2 ox + a* = x* + ox ; " 

transposing and changing sign, 3 ox = a* ; 

whence x = — = -. 

3a 3 

Solve 

4. Vaj-|-4 = 6. 11. V5^T4=V3a + 2. 

6. V7a;-38 = 6. 12. V4aj- 19 = 2VS- 1. 

6. Var^-16 = i»-2. rs. V^T5+V« = 5. 

7. Va?--l=V» — 9. -. y x — 1 

^ ^ 14- Vo; + 3 = > - ' 

8. Vaj-h3 = V3aj-l. Va?-3 

9. 5V«^^ = 3Va + 5. 15. Va; + 12 = 2+V5. 
10. \^4a; + 3 = 3. 16. V4+Va^ = a 



§411.] RADICALS. 227 

17. VaJ + 3a = 3Va. 26. a + » = VaJ^ + Sooj — 2a*. 



18. V^Ta^=6-«. o^ g^ ^__ 1 

^ f . — ~ ~~ JC — 



19. V^Ta' = V» + a. Va^-l Va^-l 

20. V« — V2=Va? — 2. gg Va;4-«+ V^ _g^ 

21. V» + 4a6=V5 + 2a. VaJ + a- V® 

22. 2 V» — Va = 2 Va? — a. 29. Var* — a^x = ^/ai^ + 6V. 

23. VS+^-V^=^^=V26. 30^ VS + 2=Vi_V^T21. 

24. Va? + a = Va + Va: — a. /- « . o 

4 31. •V^-^ + 3 = ^±^. 

25. v^+2+V^^=-7==- 1 VaJH-2 

Va;+2 

Miscbllahbous exbrcisbs. 
Simplify 

1. f 2^ V^ — i aa? V»^. 11^ Va; — 4 — 5 

-- :^^^-^^ (.+,)^ ^^^ (a. + .)V^. 

3. VaiB»-4a''ar+4a»+Vl6^. ^'g + y - 

4. (^-</|)xa/|. 13. Vll + 4 V6. 

\2__V3 \6 jg V 9 V5 + 21. 

6. Va_V6xVa+V6. ^^ V2I - 4 V6. 

7. -^±4. ,- 3 



8. 



'"+^ "" V2+V3+VB 

-wab — ax 2 

hx-Vah • V2+V3-1' 

Va; + 1+Va;-1 1 1 

9. , , 19. 1 ■— 



VSTT-Va;-1 ' V5-V2 V5+V2 

10. V^^_W^. 20. — ? -^^ — 

Va? — y + Va + y VT + I V7 — 1 



228 ALGEBRA. [§ 412. 

21. V6x</i2x</Sx</Ux^/lS0. 

22. (x-{-V^+Vf)(^-V^--Vf). 

23. (a^-\'Va + ^/b){a^—^/a--■^/b). 

24. Va^ — &^ H-(Va — ft X Va — 6) 

Knd the value of a; in 

25. -^L—+ ^ =^. 27. V^^T^+V^^ ^,. 

Vl— a;+l Vl+a;— 1 ^ ■y/a-^-x — 'Va — x 



26 






v^^ 



29. ■\/x—a—-y/x—b=Vb — Va, 30. 2 V5— V4 a?— 11=1. 

IMAGINARY NUMBERS. 

412. Since (± a)^ = + a^, (± a)* = + a^ and (± a)^= + a«" 
(§ 316), it follows that every even power of any number is 
positive; and hence the even root of a negative number is im- 
possible, and hence such a root can only be indicoUed by the 
proper sign. 

413. The indicated even root of a negative number is called 
an imaginary number. Thus, V—a and V— a^ are imaginary 
numbers. 

In distinction from imaginary numbers, all other numbers 

are called real numbers. 

An imaginary number is more properly called a non-real number, 
numbers being thus classified as real and non-real ; but the term imagi- 
nary number is in almost universal use. 

414. The indicated square root of a negative number is 
called a quadratic imaginary number. 

It may be shown by the methods of higher algebra that every imagi- 
nary number may be expressed in the terms of a quadratic imaginary 
number, and hence we shall consider herein only quadratic imaginary 
numbers. 



§417.] RADICALS. 229 

415. Since 

V^r^* = Va*x(-l)=V^xV^^=aV^ 
and V— a =Va x(— l)=Va x V— l = a^V— 1, 

it appears that any imaginary square root may be changed 
to the form aV— 1 or a'V— 1, in which a and a* are reoi 
numbers. 
The radical V— 1 is called the imaginary unit. 



Reduce to the form aV— 1 



1. V--4. 



2. V^=^. ^ 16 



• V 



-9sc« 



8. V^^256. rr^ 



^•>/?- 



«• \-^- 10. V-(a + &)*. 

416. An indicated square root of a number is squared by 
simply removing the radical sign, as ( Va)^ = a, and (V— a)^ 
= — a; and hence the square of the imaginary unit V— 1 



is — 1. 



417. The successive powers of the imaginary unit are as 
follows : 



(V^::i)8=(V3i)2xV^=T = -ixV=l = -V^^; 
(V^*=(V^^)»x(V^^ = -lx-l = + l; 

and so on. 



230 ALGEBRA. [§ 4ia 

Hence, if n denotes any positive integer, 

(V^)** =[(V=^)*]"=(-1 X -l)- = + l; 

.-. ( v=^)^+' = + 1 v^^ = v^=i: ; 

.-. (V^^)^+* = + lx-l = -l; 

418. It is thus seen that the first four powers of V— 1 are 
respectively V— 1, —1, — V— 1, and +1; and that these 
numbers occur in the same order for the powers of V— 1 
whose exponents are 4w + l to 4w + 4 inclusive. 

Addition and Subtraction. 

419. Imaginary numbers may be added or subtracted in the 
same manner as other radicals (§ 389). 



1. Add V^^ and V^^25. 

2\/^=T + byT^ = 7 V^. 

2. Simplify V- 144 - V^TIq _ V^Zie. 

V^^U4 = 12\/^T 

Simplify 

4. 3V-16-2V-25. IT ifir T 

. ^r—T ./ 9- 9- V^^-V^=4^+V^=^9^. 

7. V=^+V:^36-V=65, yi a' M a' M a«' 



§ 420.] RADICALS. 231 

Multiplication and Division. 

420. The only special difficulty in multiplying or dividing 
imaginary numbers is in the sign of the product or quotient. 
Thus, V^^ X aA^ is not V-2x(~3) = V6, as in the 
multiplication of radicals (§ 393), but V6 X (— 1)= — V6. 

This difficulty is avoided by first reducing the imaginary 
numbers to the form a V— 1, as below. 

1. Multiply V^ by V^. 

V^Te = Vo X V^^ 

>/^ X V^ = VSO X ( V^=T:)2 = V30 X ( - 1) = - V30 

2. Divide V^^^^ by V^^Ts. 

V- I2-;-V^ = V4 X l=\/4 

The same result would be reached in Example 2 by proceedinf;: as in the 
division of radicals (§ 397); thus, V372 -r- V3"3= V- 12~ (- 3) = Vi. 

A reduction of terms to the form aV— 1 is necessary in division only 
when one term is imaginary, and in multiplication when both terms are 
imaginary. 



Multiply 



3. -y/ZIs by V^r5. 7. V^ by V^=^. 

4. 5V^ by 3V^^. 8. - 3 V^=^ by 5 V^^. 

5. 12V^^ by 6V^. 9. V^^ by V^^ by V^. 

6. 8V-16 by 16V^^. 10. V^ by V^ by V^. 

Divide 

11. V=^ by V3. 14. 4.V^^ by - SV^^. 

12. V^^ by V— 5. 15. Va6 by V— a. 

13. 6V3 by 3V^^. 16. -3V^=^ by 2V-4a. 



282 ALGEBRA. [§ 421. 

421. The form a ± 6V— 1, in which a and 6 are real, is 
called the typical form of imaginary numbers. 

 

422. An imaginary number in the form a ± 6 V--T is called 
complex, and one in the form aV— 1 or V— a is called a pure 
imaginary. 

423. Two complex numbers which differ only in the sign of 
their imaginary term are said to be conjugate (§ 395). Thus, 
3— V— 2 and 3+V— 2 are conjugate, also x + y^/—l and 

« — yV— 1. 

424. Both the sum and the product of two conjugate com- 
plex numbers are real. Thus, a + 6V— 1 -{-a — 6 V— 1 = 2 a, 
and (a + 6 V^(a - b^/^^)= a^ -(- b^= a^ + ^. 

It follows that the sum of two squares can be factored by introducing 
the imaginary unit. Thus, a^ -\- b^ =(a-{- bV— l)(a — bV— 1). 

426. An imaginary number cannot equal a real number; 
for, if a? = V— a, ic^ = — a, which is impossible, since a positive 
number cannot equal a negative number. 

426. If two complex imaginary numbers are equal, their 
real parts and their imaginary parts are respectively equal ; 
for, if a-\-bV —l=X'\-y^/ —1, transposing, a— aj=(y— 6)V— 1, 
which is impossible unless a = «, and y = b. For, if a > or < 
Xf then we have a real number equal to an imaginary number 
if y > or < 6, and to it y = b, both of which are inq)ossible ; 
but, if a = Xj then y = b, otherwise we have equal to an imagi- 
nary number, which is impossible. Hence a = x, and y = b. 

427. If the denominator of a fraction is a complex imaginary 
number, the fraction may be rationalized by multiplying both 
terms by the conjugate of its denominator, as in § 400. Thus, 



l( 3-fV-2) _ 3-fV-2 _ 3+V-2 



3-V^::2 (3-V-2)(3+V^2) 9-(-2) 11 



§ 427.] RADICALS. 288 

Simplify 

17. (3 + 2 V^=^)(3 - 2 V^=^). 

18. ( V32 - 3 V^^XV^^ 4- 3 V^). 

19. a2 + 6V^^+(a + V^^)(a-V^^)- 

20. (aV^^ — cV^^)(aV^+cV^^). 

21. (V:r3+v"i:2)(V^2-V^. 

22. (i 4- 1 v^)a - 1 v^). 

23. f.-^Ya:4-.^ 



24 
25 
26 



. (V^r3^_v^r2)^(V^r3-Vir2). 



27. ^ ^ - 28. — ^ 

4 - 3 V- 2 a - aji,/^=T 

^^ 3V32 4-2V£3 , 3^ V^ + V^ . 

31. 3 2 

5-2V^^ 5 + 2V^=^ 

o„ a 4- aV— 1 a — O/'V— 1 

a — xV— 1 a 4- aV— 1 

a— V— 1 a;4-V— 1 

34. 2 v^=n: 4- 2 V^^ - 3 v^^. 

35. V^^+V^^12 4-(V^(V^2-Vi:8). 

Expand 

36. (3-V^l 39. (a4-V^/-(a-V^)2. 

37. (V^^-2V^^)l 40. (xV^^ -{- yV^y. 

38. (5V=^-V^^)'. 41. (l + VZi:)^^(l_V^)*. 



234 ALGEJBIiA. [§ 42& 



CHAPTER XIV. 
FRACTIONAL AND NEGATIVE EXPONENTS. 

428. The definition of exponent already given (§ 29) applies 
only to exponents which are positive and integral; and the 
laws of exponents established have also been considered as 
referring to exponents which are positive and integral. 

429. It remains to explain the meaning of fractional and 

2 

negative exponents, as a^ and a~^, and also to show how oper- 
ations are performed with numbers which contain such ex- 
ponents. 

FRACTIONAL EXPONENTS. 

430. It is assumed that the fundamental law of exponents, 
a** X a" = a*'*"'* (§ 84), is true, whatever may be the value of m 
and 71. 

Hence, if a"* x a*" = a"+**, then a^ x a^ = a^^^ = a ; and 
since a'^ x a^ = a, a^ = -y/a ; that is, a^ denotes the square 
root of a. 

431. Since a^ x a^ x a* = a*"*"*"*"^ = a, a^ = -J^; that is, a* 

denotes the cube root of a. 

Ill 1 1 

Since a" x a" x a** ••. to n factors =:a, a*" = -y/a] that is, a* 
denotes the nth root of cp. 



§435.] FRACTIONAL AND NEGATIVE EXPONENTS. 236 

m m IK iM fM 

~" ^ ■"• ~~Xfi ~" 

Since a" x a* x a* ••• to n factors =0** = a* a" = \^a"*; 
and hence a" denotes the nth root of the mth power of a. Hence 

432. The numerator of a fractioncU exponent denotes the power 
of a number, and its denominator denotes the root of that power. 

Thus, x^ denotes the fourth root of the cube of x, 

111 1 m M 

Since a*» x a* x a* to m factors = a* ^a"*, a"^ = (x/a)** ; and hence 

a» also denotes the mth power of the nth root of a. 

But this is strictly true only when the root is arithmetical or positive. 
For example, Vo* = ± a^ ; but (Va)* has only one value, + a*. 

433. Whatever the value of a may be, a° = 1 ; for a* -^ a* 
= a"-" = a°, and a'^-i-a'* = 1. .*. a® = 1 (§ 123). 

NEGATIVE EXPONENTS. 

434. If the formula a"* -j- a" = a""^ is true for all values of 
maiidn,then a' ^ a' = a'-' = a'^ 

But a* -5- a* = — = — • 

a* a* 

.-. a~^ = — ; that is, a~^ denotes the reciprocal of a\ 

Likewise a" -s- a^ = a*"~^"* = a""* ; 

but a* ^ a** = -7- = — 

a^"* a"* 

.*. a~** = — ; that is, a"* denotes the reciprocal of a*, 
a** 

435. It is thus seen that any factor may be transferred from 
the numerator of a fraction to its denominator , or from the 
denominator to the numerator, provided the sign of its exponent 
be changed. 

Thus a-^b-^' ^-aZ^-2-JL. «'^'- ^"V . 



286 ALGEBRA. [§435. 

Express as integers 

1- — :; — ::• 4. — : -• 7. 






a'%'^ aj-iy-^-f 



6. ""' . 8. ^y^ 



2. ^^ 6. a&^ ^ __ 

X *^ *2 

3. — _- — . 6. ^- 9. ' 

Express with positive exponents 



8«-2 



10. «*' . 


12. *■v^ 






14. 




11 «"'^"' 


13. ^^. 

ary-* 






15. 




Express w 


ith fractional exponents 
19. ^16 aV. 




22. 




16. -v/^. 


^a'h-h-'i. 


17. -y/a^T^y, 


20. ^X 


ft 1 — ; 


> 


23. 


^a«6» + VaV. 


18. ^^-27. 


21. V^-^ 


■>/d\ 




24. 


^/'a^ft^ : a-^6-*. 


Express with radical signs and positive exponents 


25. a'^ft^. 


27. SM. 


29. 






31. — i 


26. a"Vl 


28. 2cr^V. 


30. 


3_2 

4-» 




32. <^h%-\ 


Find the value of 










33. 8*. 


36. 4"*. 


39. 


64i 




42. 36-*x3». 


34. 16*. 


37. 16-^ 


40. 


lOOH 


. 


43. 25* -4- 27* 


36. 27^ 


38. 9^ X 27^. 


41. 


1000*. 


44. 9'*X36*. 



§438.] FRACTIONAL AND NEGATIVE EXPONENTS. 237 

OPERATIONS WITH FRACTIONAL AND NEGATIVE 

EXPONENTS. 

436. The several laws of exponents may be expressed by the 
following formulas : 

(1) a"* X a'' = a"*"*"*. (4) (a -5- ft)* = a"* -t- 6* 

(2) a- -s- a" = a*-". (5) (a"')" = a"'". 

(3) (aft)" = a-'ft*. (6) Var' = a'^. 

437. These formulas being true for all values of m and n 
(§ 430), it follows that the rules previously given for opera- 
tions with numbers whose exponents are positive integers, also 
apply to operations with numbers which contain fractional or 
negative exponents. 

438. Operations with radicals may be readily performed by 
substituting fractional exponents for the radical signs, and 
then proceeding as with rational numbers (§ 402). By such 
substitution the operations with radical numbers may often be 
much simplified. 

1. Multiply 5 a6^aj% by Za^b^a^y^. 

The product is found by multiplying the coefficients, and adding the ex- 
ponents of the like literal factors (§ 109). The product is 15 cfib^x^y^. 

2. Divide ofix^y^ by 3 a'aj^t/*. 

3. Multiply a + a*6* + & by a* - 6* 

a -{■ ah^ -{■ h 

- ab^ - ah - b^ 
a* -6* 



238 ALGEBRA. [§ 438. 

4. Divide 

a?* — 3 x^y~^ -\- 3 x^y^ — y"^ by x^ — 2 x^y"^ + y~^. 



x^ - 3xiy i + 3xV* - y~^ 


x^ — 2 x*y"^ + y ^ 


aji _ 2 x^y i + x^y-i 


x*^ — y ' 


- x*y"^ + 2 x*y"^ - y~i 

- Q^y~^ + 2 xiy"i - y"i 



Multiply 

6. a^ + 63 by a^ — 6*. 9. «» + a;%^ 4- y^ by a;^ — y^. 

6. a* + 6""* by a* — 6"*. 10. a;"* + »"* + ! by. a;"* — 1. 

7. aj + «"^ by a; 4- «"^ H. a + a;* + 2 by a; + a;* — 2. 

8. x^-\-y'^ by x~^-^y^. 12. aj*+a*+l by a?-*— ar*+l. 

Divide 

13. a-h by a* + 6*. 16. 27 x-^ ^^y'^ by 3aj"* -f 2y\ 

14. a H- 6 by a^ + 6^. 16. a;^ — y^ by a;^ + 2^». 

17. a* — a6* + «*& - &* by a* — 6^ 

18. a; + a?^2/^ + y by a?^ — a;^y* + y^ 

Extract the square root of 

19. 4a — 4aMH-&. 21. a?^ — 4a;* 4-2a; + 4a;^ + a;*. 

20. aj-^+2a;"'V*+y"* 22. a;-'*— a;-V^4- ^^"^" — a?"V"^4-y-^. 

4 

Simplify 23. (a3)-2x(a-^^. 26. A/(aW^^^W^. 

24. (3- 2 + 1)* 27- («V)- X (pj:)^. 



1 1 -(w-l) 



§442.] QUADRATIC EQUATIONS. 289 



CHAPTER XV. 
QUADRATIC EQUATIONS. 

439. An equation which, when cleared of fractions, contains 
the square of the unknown number, but no higher power, 
is an equation of the second degree. An equation of the second 
degree is called a quadratic equation. Thus, 4a^ = 16, and 
3 a^ — 4 a? = 15, are quadratic equations. 

440. A quadratic equation may be reduced to the general 
form aa? -{-hx-{-c = 0, in which a, 6, and c are known numbers. 
The known numbers, a, 6, are called the coefficients of the 
equation ; and the term c, the constant term. 

The term c may also be made a coefficient by writing the equation in 
the form of ax^ + &x + cico = 0. 

441. When the coefficient h is zero, the equation becomes 
as? + c = 0, and is said to be incomplete, because the first power 
of X is wanting. When no one of the coefficients is zero, the 
equation is said to be complete. Hence 

An incomplete quadratic equation contains only the square of 
the unknown number. 

A complete quadratic equation contains both the square and 
the first power of the unknown number. 

An incomplete quadratic equation is also called a pure quadratic ; and 
a complete quadratic equation, an affected quadratic. 

442. The root of an equation is the value of the unknown 
number in it (§ 138). When a root is substituted for the 
unknown number, it satisfies the equation; that is, reduces it to 
an identity. 



240 ALGEBRA. [§ 443. 



INCOMPLETE QUADRATICS. 

443. An incomplete quadratic equation is reduced, if neces- 
sary, to the general form as? = c by the same transformatioDs 
as those employed in the solution of simple equations. 

444. An equation of the form aa? = c is solved by dividing 
both terms by a, and then extracting the square root of both 
members. 



Thus, 


aa? = c; 


dividing by a, 


a 


extracting square root, 


x = ± 



445. The substitution of +\l- or — %/- for x will satisfy 

the equation (§ 442) ; and hence an inwmplete qxiadratic equation 
has two rootSf numerically equal, but having opposite signs. 

446. A root which can be exactly found, as V9, is a rational 
root. 

A root which can only be found approximately, as V5, is an 
irratiortal root, or surd (§ 371). 

A root which cannot be found either exactly or approxi- 
mately, as V— 9, is an imaginary root (§ 413). 

1 . Solve the equation ^±-? + ^^ = ??. 

^ aj-3 aj + 3 10 

Clearing of fractions, 10 (x + 3)2 + 10(x -Sy = 29(x^ - 9) ; 

expanding, 10x2 + 00 a; + 90 + lOx^ - 60a; + 90 = 29x2 - 261 ; 

transposing and simplifying, x2 = 49 ; 

extracting square root, x = ±7. 



§446.] QUADRATIC EQUATIONS. 241 

2. Solve the equation ^^^=2aj2- 7. 

Clearing of fractions, 6^2 _ 5 _ 4 3.2 _ 14 . 

transposing, etc., x^ = — 9; 

whence x = ±^/^^ = ±SV^ri. 

Since the square root of a negative number is imaginary (§413), the 
value of X can only be indicated. 



3. Solve the equation x — ^^ — 3 = 



V?^-3 



Clearing of fractions, xy/x^ — 3 - (a;2 _ 3) _ 2 ; 

transposing and uniting, xy/x^ — 3 — x^ — 1 ; 

squaring each member, a^ — 3a;2 = a;4 _ 2 x^ + 1 ; 

transposing and simplifying, x^ = — \\ 

whence x-=± V^^. 

Solve the equations 

4. 7a? = mi. ^^ a;4-2 x-2 ^^ 

6. 11 ic*- 9 = 35. ' x~2 x + 2 



4 



X 



7. i^-5 = 7. ^^ 



16. ^+ 1 1 



4 a; — 3 

10. -^-=10. 2aj2^ 4a2 

11. x-\--=2x. 19- — 



2a; x-\-2 



3aj aj4-5 o 

13 ^-4 ^ ar^4-ll 01 a?4-5 a?-5 _ 15 

3 4* • aj-5"^aj + 5"" 4* 
white's alo. — 16 



242 ALGEBRA. [§ 446. 

22. VT+^-x l 25. ?1^2L=a^, 

a* 4 ax 

23. -^2- = — !?_. 26. ^±60^^_^jzl2. 
or — n jf — m 6 3 

^*« Va-i-a V 27. — ' = — 

■ya — x x — a x + a c 

a? 4- 6 a? — 6 

29. ^ ^'^ 1 



30. 






31. Two numbers are to each other as 3 to 5, and the differ- 
ence of their squares is 256. What are the numbers ? 

Suggestion. Let 3 x and 6 a; be the numbers. 

32. Two numbers are to each other as 2 to 7, and their 
product is 126. What are the numbers? 

33. A rectangular field contains 6 acres, and its sides are 
to each other as 3 to 4. What is the length of each side ? 

34. The length of a rectangular field is 2^ times the width, 
and the field contains 9 acres. What is the length of each 
side? 

35. The sum of two numbers is 16, and their product is 60. 
What are the numbers ? 

Suggestion. Let 8 + x and 8 — a; be the numbers. 

36. A father's age is to his son's age as 5 to 2, and the 
product of their ages is 640. What are their ages ? 

37. Two numbers are to each other as a to &, and their 
product is c^. What are the numbers? 

38. A son's age is to his father's age as a to 6, and the 
product of their ages is m. What are their ages? 



§449.] QUADRATIC EQUATIONS. 248 

COMPLETE QUADRATICS. 

447. There are, as will be seen, several methods of solving 
complete quadratic equations; and it is advisable for pupils, 
especially those who expect to enter higher institutions, to 
acquire a knowledge of those in more common use, though 
in practice it may not be best to use more than two of them. 

Method op Solution by Factoring. 

448. A product is zero if any one of its factors is zero 
(§ 22) ; and, in order that any product may he zero, at least one 
of Us factors must be zero. 

For example, if (x — 2) (a; 4- 3) = 0, then a5 — 2 = 0, or 
a? 4- 3 = 0. But if a; - 2 = 0, a; = 2; and if a? -f 3 = 0, a = - 3; 
and, since either value of x thus found will satisfy the equation 
(x — 2)(x -f- 3)= 0, its roots are 2 and — 3. 

449. It is thus seen that any quadratic equation whose first 
member is a product, and whose second member is zero, may 
be solved by equating each of its two factors to zero, and solving 
the resulting equations. 

It is here assumed that the factors are finite. 

1. What are the roots of the equation (x + 3)(a: — 1) = ? 

Equating the factors to zero, a: + 3 = 0, and x — 1 = ; 
whence x = — 3, and x = 1. 

Hence the roots are — 3 and 1. 

Find the roots of 

2. (a;-7)(aj-2)=0. 6. (« + i)(aJ- i)=0. 

3. (aj + l)(aj-5)=0. 7. (a; - 6)(a: -h c) = 0. 

4. (a?-6)(a;-4)=0. 8. (a - a)(aj- V6)=0. 

5. (a? - i)(aj + 2)= 0. 9. (aj+V^)(aj-Va)=0. 



244 ALGEBRA. [§ 450. 

450. A quadratic equation of the form a? -\- bx -\- c = may 
be readily factored by § 202, and its roots thus found, pro- 
vided c is the product of two rational factors whose algebraic 
simi is b. 

For example, let aj* + 3a; — 10 = 0. 

Factoring by § 202, (x + 5)(x - 2) = ; 

equating factors to zero, a? + 5 = 0, and « — 2 = ; 
whence x = — 5i and x = 2. 

Hence the roots of the equation are — 6 and 2. 

10. Solve the equation a^ — 7x = — 12. 

Transposing - 12, a;^ - 7x + 12 = 

factoring, (a; — 4) (x — 3) =^ 

equating to zero, oj — 4 = 0, and x — 3 = 

whence x = 4, and x = 3. 

Hence the roots are 4 and 3. 

451. A quadratic equation of the form oo* + fta? + c = may 
be factored by § 205 or by the general formula, § 208, and its 
roots be thus found. 

11 . Solve the equation 3 a* + 7 « = 6. 

Transposing 6, 3x2 + 7x - 6 = 

factoring by § 206, (3 x - 2) (x + 3) = 

equating to zero, 3x — 2 = 0, and x + 3 = 
whence « = f » and x = — 3. 

Hence the roots are f and — 3. 

Solve by factoring the equations 

12. a^-aj= 12. 18. a^- 10a; = 56. 

13. a^-4:x = 4:5. 19. o^ -f 15 a? = - 26. 

14. a^-10x=z-21. 20. ar» - 15 a? = 154. 

15. ar* - 12 aj = - 32. 21. 2a*+6a? = 20. 

16. a:* + 3aj = 28. 22. 3a*- 7a? = 40. 

17. ar^ + a = 56. 23. 5a*+27a?=18. 



§454.] QUADRATIC EQUATIONS. 245 

24. 6a?-7aj = 20. 28. a? + 2a^ -^Sax = 0. 

25. 4ar*— lla: = — 7. o« «2 / b\ , , 

26. 4ar^-23a; = -15. 



27. aj^ + aaj + 6a? = — a6. 30. aj^+(Va4-V6)aJ= — Va5. 



452. ^ny complete quadratic equation may be solved by 
factoring, but the method has no special advantage when the 
factors cannot be determined by inspection. When the factors 
can be thus readily determined, no other method of solution 
need be used. 

453. When a quadratic equation can be readily factored by 
inspection, its roots may be written at once without equating 
the factors to zero. 

Thus, the roots of aj^ — 2a; — 24 = are seen to be — 4 and 6. 

Write, without equating factors, the roots of 

31. ar* + 6 a; -h 9 = 0. 34. ar^- 7 a; -60 = 0. 

32. ar^- 5 a; 4-6 = 0. 35. 2ar^ + 10a; + 8 = 0. 

33. a;2_^3a._10 = 0. 36. 3 a;^- 12 a; -15 = 0. 

454. Conversely, if the two roots of a complete quadratic 
are given, the equation can be formed by findiiig the product of 
the two binxymial factors whose first terms are x, and whose second 
terms are respectively thei;wo given roots with signs changed, and 
then writing the product thus found equal to zero. 

Thus, if —2 and 3 are the two roots of a complete quadratic, 
the equation is (x -f 2)(x — 3) = 0, or a;* — a; — 6 = 0. 

It can be shown that a;^ — a; — 6 = is the only equation whose roots 
are — 2 and 3 with no other roots, but the proof is too difficult for 
insertion here. 3a;2_3a; — 18 = is reducible to x^ — x — Q^Q^ and 
hence is no exception. 

37. Form the quadratic equation whose roots are 3 and —5. 

a; = 3, and » = — 6. 
Transposing 3 and — 5, oc — 3 = 0, or« + 5 = 0; 
whence (x — 3) (a; + 5) = ; 

/. «3 + 2 a; - 16 = 0. 



246 ALGEBRA. [§ 455. 

38. Form the quadratic whose roots are f and —J. 

as = j, and x = — J. 
Transposing, oc — f = 0, and as + J = 

whence (« — i) (« + i) = <^ 

... a;2-ia;-i=0 
clearing of fractions, 6 x^ — a; — 2 = 0. 

455. The required equation may be written at once by 
changing the signs of the given roots, and then making their 
algebraic sum the coefficient of aj, and their product the third 
term. 

Thus, if the two given roots are a and , the equation is 

b 



«^+(-«+?>-f=«- 



Form the quadratic equation whose roots are 

39. 1 and — 5. 44. — 4 and 0. 

40. — 3 and — 4. 45. —VE and VS. 

41. — 2 and ^. 46. V3 and — V3. 

42. I and— J. 47. 2 — V3 and 2 + VS. 

43. — i and — 4. 48. - and — a. 

8 2 . ^ 

Solution by Completing the Square. 

456. Any complete quadratic equation may be reduced to 
the form of ±hx = c by dividing, if necessary, both members 
by the coefficient of a^, 

457- If the square of ^ of 6, or ( - ) , be added to both mem- 
bers of a^ + bx = c (1), called completing the square, the equa- 
tion becomes 

The first member of (2) is now a perfect square, the ex- 
tremes being each a perfect square, and the middle term twice 
the product of their square roots (§ 186). 



§457.] QUADRATIC EQUATIONS. 247 

Extracting the square root of each member of (2), we have 



"■2 



and transposing + -> a? = — - ± -v/c + — 
It is thus seen that the equation has two roots, 



- 1 +>K»^ -!->/< 

It follows from § 454 that the first member of the equation x^-\-bx — e 



= is the product of x + ^- -}-Jc + —\ and ac + /- --Jc + — V 

1. Solve tte quadratic 3 aj* — 12 a; = 15. 

Dividing by 3, x^ — ^x = 6; 

completing the square, a;^ — 4x + (2) 2 = 6 + 4 = 9; 
extracting the square root, aj — 2 = ± 3 ; 

whence x = 2±3 = 5 or — 1. 

2. Solve the quadratic a* 4- 11 a; = 33|. 

Completing the square, x^ + n a; _^ (i^)2 = iji + iji = aja. 
extracting square root, « + ^ = ± r^ ; 

whence x = - J^ ± ^ = 2} or - 13J. 

3. Solve the quadratic ^^Lzl _ i^±2 = 2 aj - 3. 

X 5 

Clearing of fractions, etc., — 14 x^ + 28 x = 35 ; 

dividing by - 14, x^ - 2x = - f ; 

completing the square, x3-2x + l=-f + l=-}; 

extracting the square root, x — 1 = ± V— f ; 

... x = l±v^ = l±i\Ar6. 

The two roots are imaginary, as they will always be when c + — is 
negative (§ 413). ^ 



248 ALGEBRA. £§ 458. 

Solve by completing the square 
4. a^-8a; = -15. 7. iB*-12a; = 45. 

6. aj» + 12a; = -20. 8. jb* - 20 a? + 19 = 0. 

6. «»-f 4a: = 21. 9. a^ + 18 a? - 88 = 0. 

10. 2a^-10a? + 6 = lla?-»*-30. 

11. 3x^-llx = 20. 

12. 5a:»-aj + 19 = 3x» + 15«-15. 

13. 7a^-12aj = 580. 

14. (3a;4-10)(3a;4-7)=2(2-2a;-a^. 

15. 2ic2 + 94a;4-420 = 0. 

16. 3aj2 - 52a? 4-118 =(5- 2a;)(3a; + 2). 

17. 2(a;-3)(a:-4)=aj*~25. 

-^ aj-l-32 39 ,^ 5a; + 3^7a;-f2 



ar^-2a;-20 2a;4-l 8a;-4 lla?-3 

20. ^+7=?. 21. T^ + ^-^^ = i^. 

2ar X 4 4 2 

22. 2(a: - 2)(a: - 3) = (a? -4)(aj- 3) +10. 

23. 35-\(a^-\-50)=a^-^i(a^-10). 

458. To solve a complete quadratic by completing the 
square, 

Reduce the equation to the form a^ -^bx=c. 

Add the sqtiare of one half the coefficient of xto both membersy 
thus completing the square of the first member. 

Extract the square root of each member of the resulting eqiLOiion, 
and then find the two values of x. 

The problem of solving a complete quadratic equation is thus reduced 
to that of solving two simple equations. 



§459.] QUADRATIC EQUATIONS. 249 

Other Methods op Completing the Square. 

Note. The four sections following (§§ 469-462) may be omitted by 
beginners. 

459. Instead of dividing both members of the equation by 
the coefficient of aj^, it is sometimes more convenient to multiply 
both members, if necessary, by siich a number as mil make the 
coefficient of x^ a perfect square, and then complete the square 
by adding to ea^h member the square of the quotient obtained by 
dividing the coefficient of x by turice the square root of the coeffir 
dent of x^. 

Take, for example, the equation 3 oj^ — 10 a; = — 3. 
Multiplying both members by 3, we have, 

9iB2-30a? = -9; 

(30 \^ 
o X ^J 

9aj2_30a;4-52 = -9 + 25 = 16; 
extracting the square root, 3 a; — 5 = ± 4 ; 

transposing and uniting, 3a5 = 5±4 = 9 or 1; 

.-. a? = 3 or \, 

When the coefficient of a? is already a perfect square, the 
square may be at once completed as above. 

1. Solve the equation 18 a* — 15 a; = 42. 

Multiplying both members by 2, 36 x^ _ 30 a; = 84 ; 

completing the square by adding (f ?)^ =(l)^» 

36x2 _ 30X +(1)2 = 84 +(f)2= 161 ; 

extracting the square root, 6 x — f = db V" ; 

transposing and uniting, 6 x = f ± y = ^^ or — 7 ; 

.'. x = 2 or — 1|. 

Solve as above the equations 

2. 8aj2~12aj = 80. 4. 10 ar^ + 2 = - 12aj. 

3. 3aj2-8aj + 4 = 0. 6. Ta^ _ 28a; = - 21. 



260 ALGEBRA. [§ 460. 

6. 9iB" + 9a5 + 2 = 0. 10. S2 a^ - 60 x = 272. 

7. 27fiB*-30ir = 48. 11. 25 aj = 6 aj* -f 21. 

8. 5a*~44aj = 9. 12. 60 ic* - 27 = 15 a?. 

9. 12 aj* = 24 a? + 420. 13. 18a^- 27aj- 26 = 0. 

460. The foregoing methods of completing the square involve 
fractions when twice the square root of the coefficient of x^ is 
not a factor of the coefficient of x ; and these fractions some- 
times necessitate much work, as will be seen by solving by 
either method the equation 29 a^ — 31 a; = 54. 

461 . The first member of any complete quadratic equation 
may be made a complete square without fractions. 

For, take an equation of the general form 

tta^-\-bx = c, (1) 

Multiplying both members by 4 a (4 times the coefficient of 

a^), we have 

4aV + 4a6a; = 4ac. (2) 

Since 4 a6 -5- 2 V4 a* = b, the square of the first member of 
(2) is completed by adding b^ (the square of the coefficient 
of X in (1), the given equation) to both members ; thus, 

4 a^a^ + 4 a^>a; + &^ = 4 oc + 6». (3) 

Extracting the square root, we have 

2aaj + 6 = ±V4acTW. (4) 

Solving the simple equation (4), 



— 6 ± V4 ac + &* 
X = • 

2a 

462. Hence any complete quadratic equation may be solved 
without fractions as follows : 

Multiply each member by 4 tim£s the coefficient of x*. 
Add to each member of the resulting equation the square of the 
ooeffi^dent of x in the given equation. 



§463.] QUADRATIC EQUATIONS. 251 

Extract the square root of both members of this equation^ and 
solve the resulting simple equation. 

This method was first used by a Hindoo mathematiciaii, and for this 
reason it is known as the Hindoo method, 

14. Solve the equation 13 a/^ — 15 a? = 22. 

Multiplying by 4.13, 4.13'ia;2 _ 4.13.16a; = 4.13-22 = 1144 ; 

completing the square by adding 15^, 

4.132x2 _ 4.13.16X + 162 = 1144 _(. 152 = 1369; 
extracting square root, 26x — 16 = ± 37; 

transposing, etc., 26 x = 62 or — 22 ; 

.*. X = 2 or — \l. 

When the coefficient of x in the giyen equation is an even number, 
the square of the first member may be completed without fractions by 
multiplying both members by the coefficient of x2, and then adding to each 
member the square of half the coefficient ofx in the given equation. 

Solve the quadratic equations 

15. 3a^4-2a; = 2|. 20. 15x^-Sx + l = 0. 

16. I3a^-24aj = 205. 21. 20aj«- 54a?= 104. 

17. 15a^-207 = 24a?. 22. 11 a^ - 24 = 10 a?. 

18. -10a^-f-23a; = 12. 23. 3 aj» + 2 a? = 56. 

19. 21a* + l = -10aj. 24. 5a^-S = 6x. 

Method op Solution by Formula. 

463. The solution of a complete quadratic equation of the 
general form aa?-\-bx=c gives as the two roots 

— b±V¥~+Tac 
x = • 

2a 

Instead of working out the solution of every equation from 
the beginning, completing the square, etc., we may write out 
the roots at once by substituting for a, b, and c in the above 
general formula their values in the particular equalion, as 
shown below. 



252 ALGEBRA. [§ 464. 

1. Solve Uie equation as* — 11 a; = — 24. 

In this case a = 1, 6 = — 11, and c = — 24. 

Sabstitating these values of a, 6, and r, in the formnla, we have 

_ 11±VTP"H-4(-24) _ 11 j:Vi2n^^96 _ ll ±V25 . 
*~ 2 2 2 ' 

2 

In snbstitnting, special attention most be given to the sigjis of the co- 
efficients. 

2. Solve the equation 17 ic* + 8 a; = 21. 

a = 17, 6 = 8, and c = 21. 

Substituting these values in the formula, 

^ -8±V64+1428 -8±Vl492 
34 34 

^ ^ - 8 ^38.6264+ ^ ^^ ^^ -1.3713. 
34 

Since V1492 is an irrational number, the two roots of x are surds. 

464. The correctness of the roots obtained may be verified 
by substituting them for x in the given equation; but it is 
usually better to verify by the aid of the principle stated 
in § 466, that the sum of the two roots in the general for- 
mula = ^^^, and their product = 



a a 

Thus, in Example 2, above, -84-Vn92 -8-Vi492^-8 
' ^ ' ' 34 34 17' 

and -84-Vn92^8.^Vi492^--21 
34 34 17 

In using this method, a glance at the coefficients of the equation will 
usually show the correctness of the results. 

Solve by formula and verify the roots of 

3. aj2 - 24 0? = 481. 6. ar^ - 24 a? = - 119. 

4. ar^- 41a? -348 = 0. 6. 3ar^-lla? = 4. 



§466.] QUADRATIC EQUATIONS. 263 

7. 2ar2 + 38aj = 364. 11. 3a? -llx-20 = 0. 

8. 4aj2-17a; = 42. 12. -2x'^4:X-5 = 0. 

9. 5x2 + 21 a; = 62. ^3 rnoi? -{- nx — 4: = 0. 
10. 7x2^34^.^24 = 0. 14. mV + wx-l = 0. 

466. Assuming that the coefficients in the general equation 
ox* -\- bx = c are finite, and not zero, it can be shown, by 
methods that properly belong to higher algebra, that it has 
two roots, and only two. 

466. The roots of a quadratic equation have the following 
general properties: 

I. Adding the two roots as expressed in the formula 

^ — =— — — — 2^, it is found that their sum is -^^^^; and, multi- 
2a a ' ' 

plying them, it is found that their product is , 

II. If 6* + 4 ac be positive, -y/b^ -f 4 oc is real, and the 
roots of the equation are real and unequal. 

III. If 6* 4- 4 ac be a perfect square, the two roots are 
ra>tional ; if 6* -f 4 ac be not a perfect square, the two roots are 
irraJtioiud. 

IV. If 6^ + 4 ac is negative, both roots are imo/ginary. 

V. If 52 + 4 ac = 0, both roots reduce to •^— , and are thus 

2a 

equal; and it is then said that the equation has two equal roots, 

LITERAL QUADRATICS. 

1. Solve the equation (a — x) (a + x) = a (x — a). 

Multiplying, transposing, etc., x^ + ax — 2 a^ = 
factoring, (x — a)(x + 2 o) = 

equating factors, x — a = 0, and x + 2 a = 

. *. X = a or — 2 a. 



ALGEBRA. [§ 40$. 

2. Solye tiie equation 2* + m' ~ nx = »ii. 

Complelmg the aqoaie, a*+ (»—■)*+ (^'^i^ V=**+ *^^^*'* 



4 • 



cxtiaeliDg the square root, ^^^ m * =-^.J*±J!; 

2 2 



x = -?ill«±"*^ 



2 2 

.-. x = n or — m. 

3. Solye by fommla the equation €12?— (a* — 5)x = a&. 

2a 2a 

.-. x = =a,o>r = • 

2a 2a a 

Solve the following equations by any one of the above 
methods: 

4. a? — OiX — x = — a. ab 

. ^ 11. 2 + ^ = a + 6. 

^ff 12. x-4a6 = (^ + ^X^-^) 

7. aaj«-2aaj + 2 = 2. 13. ^ 5L_ = _«5_. 

a + a5 a — x a — x 

8 £±£ + — 2L-_1 = 

a a — x ' 14. a* («* + «*) = 2 rfoj+l. 

9. a?-(n + l)x = -n. ^g aj»-2aaj + te-2a& = 0. 

in «^ g' 

^"' ^TT~a + l' ^®' ^— (« — & — c)a = ac + a6. 

17. aj»-(2c + 2(f)aj = 4c(i-3(?-(P. 

4a& 



18. (a + 6)aj* — 2(a — 6)aj = 
19. 



a + b 
x-\-l m -f 1 



Va Vm 



§467.] QUADRATIC EQUATIONS. » 266 



HISCBLLANBOUS EZBRCISBS. 

467. Solve the following quadratic equations, each by the 
method that may seem best adapted to its solution. Verify 

the results by seeing whether the sum of the two roots is ^^^^, 

— c ^ 

and their product (§ 464); or by direct substitution. 

a 

19. -H =a4-6. 

2. a^-aj-182 = 0. »-^ «-« 

3. 2««-3aj-6 = 0. 20. ? + ^ = --|-l 

« 9 o M a X a b 

4. 3a^-8aj = -4. 

6. 9aj*-12a; + 4 = 0. 



7. 12aj2-26a; = -13. 

8. .i^-h2a^ = 3aa;. 



21. a^ + a« — 6aj — a6 = 0. 



6. 6a^ + 7a; = 160. 22. aj*^ ca?- — + — = 0. 



a a 
23. a^-2(a-6)aj-f &* = 2a6. 



9. 2a^ + 3a. = 2. ^4. --_-__ = l. 

10. 12ar* — » — 1 = 0. a- a 

25. -^ + -±-- = 3. 

11. 0^ + 15 + 1 = 0. ^-^ ^ + ^ 

^ ^ 2g _i 1 _ 3 + a?' 

12. iB*4-4aj = 0. * 6 -a; 6-f-a? ft^-aj* 

13 2^4.5-1 = ^"^^ Q^-2ax + 12x = 24:a, 

6 3 «« « . 2a 3 

28. -H = -. 

14. a^_?-5 = 0. a 0. a 

2 1 IS 

29. aj-hi = i?- 

15. a;-|-- = a4--- 

« » 30. 6aj(aj-3)=-2^-12a?. 

16. _?- + _iL.= 4. 31 3^±4__30-2^^7£~14, 
a + x a-x 5 a.__5 10 

17. ba?-(a-\rab)x = -a\ iq U-2x 5 

32. — — —  = — • 

IS. a?-2aa = b-a\ x a? 2 



2S6 ALGEBRA. [§ 46{. 

33. ?±i-i£±I = 12z:^. 47. 2aj-5VS = 3. 

^ ^ ^-^ 48. 2a;-fl=V6a; + 3. 

34. ?^ -j- —L- = a? -h 11. 49. aj 4- 2 V3a; -f- 1 = 0. 

5 aj — 4 



60. ■y/a^^-{-Vb—x=- 



b 



35. 10(x -h 2) = ^^. ' '" \^J_I V6^ 

* 61. V5 + V5x-f 1 = 2. 

36. ^ + ^ = L 52. aj + 2V^ir5 = 5. 
a — 2 a; — 3 2 

37. 3(a?+4)24-2(a;-17)*=274. ^^' Vx + 1+— ==2. 

38. aj* + 4naj=:4a:* + l. o 

39. a^-a^ = a-x. ^^' Va. + l-hVS = 

40. a^-7aaj = 78al 

41. -^-H-5±i5 = l ""' ~V^"^ 
a: + a a: 2 

1 66. Vx — a^ ^ 



V« + l 

55. ^±-1 — 5-±l. 



42. aj = n. Va; + 2a 



a; 



I, 57. 2Va^-64 = 2aj-8. 

43. aa;4-- = c. , _ 

X 58. vaj + 5— Vx = l. 

44. (m+w)ar^ 'HJlH.xz^m+n, 59. V2 a; + 4=^/5 4. 6 4-1 

'^ m— n \2 

45. a^-n^ = m(2a:-m). ^^ V2^+l= ^ + ^ +1, 

46. c^ar^ - acic = ^(6 - a^). V2aj + 1 

61. (5a;-2)2-(3aj + 2)« = (a;-3)2-l. 

62. 1 1(^ + 3) , 4(0^^5) , 78 



aj-f4: aj-4 a^-16 

63. (a^ - 6>-2 _ 2(a2 4. 52)a; = 52 _ ^2^ 

64. V2a; + 9-hV3a;-15=V7aj-h8. 

65. 4=-f ^ = ?. 

a;+V2-ar^ a;-V2^ar^ ^ 

66. A/l + Vi = Vl4-2aj. 



§470.] QUADRATIC EQUATIONS. 267 

EQUATIONS IN THE QUADRATIC FORM. 

468. Any equation is said to be quadratic in form when it 
is composed of three terms, two of which contain the unknown 
number with an exponent in one term twice its exponent in 
the other. Thus, the equation 02;^ + 62?" = c is quadratic 
in form. 

469. Any equation reducible to the form 025*" + 62;" = c (1), 
in which z represents any expression simple or compound, and 
n any exponent, may be solved, at least in part, by the methods 
for the solution of complete quadratics : for, denoting z* by 
«, and hence z^ by a?, (1) becomes 

aa? -\-hx = c'^ 

whence a. or ^ = -^±^J>' + ^<^. 

2a 

Whether the given equation can be completely solved will 
depend upon the possible solution of the two equations 



"^^ Y'a ' 



2f = 



2a 



470. The methods of procedure in solving such equations 
will be best understood from the following examples. 

1. Solve the equation aj* -f- 21 ar* = 100. 
Treating 7? as the unknown number, we obtain 



x^ = - lOJ ± V( V-)2 + 100. 
.-. a;2 = _ioi ±i4i=4or -26. 

Solving the equations, 

aj2 = 4, a; = ± 2. 

a;2 = _ 25, X = i 5 V^. 
white's alo. — 17 



268 ALGEBRA. t§ 470. 

It is thus seen that the given equation has fonr roots, two being real, 
and two imaginary; and it may be found by substituting that each 
root will satisfy the given equation. 

2. Solve the equation 3ofi + 20a? = 32. 
Treating ofi as the unknown number, we obtain 

^^ -10±Vi00T96 ^-JL0^U^4^^_e 
3 is 8 

Extracting the cube root, x = V^ or — 2. 

This equation has also four imaginary roots, the finding and explaining 
of which belong to a more advanced treatise. 

8. Solve the equation 5^/x — S-s/x = U, 
This equation may be presented under the form 

Treating x* as the unknown number, we obtain 

x* = ^ ^ = 2 -^ 2 --2 or — y 

o 5 o 

Hence x = (a;*)* = 2* or (- J)* = 16 or 3i^. 

4. Solve the equation Va; + 4 — 5^a; + 4 = — 6. 

Changing to (05 + 4)2 — 6 (as + 4)* = — 6, and treating (a; + 4)* as the 
unknown number, we obtain 

(x + 4)i = t ± V(i)2 -6=|±i = 3or2. 

Hence a; + 4 = [(x + 4)*]* = 3* or 2* = 81 or 16. 

/. a; = 77 or 12. 

5. Solve the equation a?=zl. 

Transposing, x* — 1 = ; 

factoring, (a; - 1) (x^ + a; + 1) = ; 

equating factors to zero, x - 1 = 0, .-. x = 1 ; (1) 

x2 ^. a; + 1 = 0. (2) 



§472.] QUADRATIC EQUATIONS. 269 

Solving (2) BS a quadratic, x = ^ — ^ "" — 

Hence the equation x^ = l has three roots, 1, ^ — ^ — ^^-^y and 
— = — , one root being real, and the other two imaginary. 



471. Certain equations which are not quadratic in form may 
be put in the quadratic form by eliminating a factor. 

Solve the equations 
6. 0^-100^ + 9 = 0. 19, a?-aj* = 56. 

20. a;-fo=Va + o + 6. 

8. aJ*-6a^-4 = 12. ^ ^^ , ^^^ 

« « , .o 21. a^-20a» = 189. 

9. a^-2a^ = 48. 

10. 0^-4 a:«- 28 = 4. ^2. a^-7a^ +Va^-7ar -f 18=24. 

11. o^ — 2po^=:q. 23. ic^+eVif^— 2aj+5=2aj-|-ll. 

12. aj*-3aj2 = 2VaJ*-3a?. 24. 9aj+Var^-3aj+5=3aj2+ll. 

13. 7aj«-12a^ + 5 = 0. 25. VS + S + ^t^^T3 = 6. 

a? 



14. a; 4- V5aj 4- 10 = 8. 

15. x+Vl0x-\-6 = 9. 26. 2(Vr=^+l)= 

16. (a^ + 2)2 = 2aJ* + 8. n 

i.y / — T^ . / — T^ o r 27. af — 2cKC2 = y. 

17. Vaj + 6+V« + 3 = 3Va:. 

18. Va + 2 = 2V2aj + l. 28. 3aj*-|aj* = 38. 



VT+^-1 



29. Va* — a^ + 05 Va^ — 1 = a VI — aA 



^o ^ ~ ^^ ax — o^ X 



a + V2aaj-a^ a — « 

472. Equations of the third and higher degrees may be 
readily solved by factoring^ if, when the second member is 
made zero, the first member is the product of rational factors. 

Take, for example, the equation ic* — 7aj — 6 = 0, 



260 ALGEBRA. [§ 472. 

Factoring by synthetic division (§ 215), 

equating factors to zero, aj + l=0, aj-h2 = 0, a? — 3 = 0; 

whence x = — l, 05 = — 2, x = 3. 

Hence the roots are — 1, — 2, and 3. 

Solve by factoring by synthetic division 

31. o^-lOaj* -♦-31a;- 30 = 0. 33. a*-5a^-f4 = 0. 

32. a^-3i?-Sx + 12 = 0. 34. aJ*-4«»-7a^-f 34aj-24=0. 

PROBLEMS INVOLVING QUADRATICS. 

1. The perimeter of a rectangular field is 600 yards, and its 
area 14,400 square yards. What is the length of the sides ? 

600 yd. -T- 2 = 250 yd., length of two adjacent sides. 
Let X = length of one side ; 

then 250 — x = length of the other side. 

Hence, by the conditions, 

x(250 -x)= 14400 ; 
whence x^-250x = - 14400. 

Solving equation, « = 125 ± 35 = 160 or 90. 

If we take x = 160, the other side is 90, and vice versa. Thus, though 
X has two values, the problem has but one solution. 

2. By selling a lot of goods for $ 24, a merchant lost as 
many cents on the dollar as he paid dollars for the goods. 
How much did he pay? 

Let X = number of dollars paid ; 

then a; — 24 = number of dollars lost. 

Hence, by the conditions, x x x = (x — 24) 100. 

Simplifying, etc., x^ _ iqo x = - 2400 ; 
solving equation, x = 60 or 40. 

Either of these values of x satisfies the conditions of the problem : for, 
if he paid $60 for the goods, he lost 60% by selling them for $24 ; and, il 
he paid $ 40 for the goods, he lost 40 % by selling them for 1 24. It is thus 
seen that the problem admits of two solutions. 



§473.] QUADRATIC EQUATIONS. 261 

3. A drover bought a number of oxen for $ 400; but, if he 
had obtained 4 more for the same money, the price paid per 
head would have been f 5 less. How many oxen did he buy ? 

Let X = number bought ; 

then 452 = price per head, 

X 

400 
and = price per head if 4 more had been bought. 

05+4 

Hence, by the conditions, 
400 ^5^400 



x + 4 X 

Solving equation, « = 16 or — 20. 

The negative root (—20) is not admissible, since it will not satisfy 
the conditions of the problem, and hence the number of oxen bought 
was 16. 

4. The sum of the ages of a father and son is 65 years, and 
the product of their ages is 250 more than 10 times their sum. 
What is the age of each ? 

Let X = father's age ; 

then 66 — a; = son's age. 

Hence (65 - x) x = 900. 

Solving equation, x = 45 or 20. 

The father's age is 45 years, and the son's 20 years. Here the second 
value of X is inadmissible, since it would make the son older than his 
father. 

6. Divide the number 12 into two parts such that their 
product will be 40. 

Let X = one part ; 

then 12 — X = the other part 

Hence (12 - x) x = 40. 

Solving the equation, x = 6 ± 2V— 1. 

Both values of x are imaginary, and hence the given problem is 
impossible. 

473. It is shown by the above examples that the solution of 
problems involving quadratics may give results all of which do 
not satisfy the conditions of the problem. This is due to the 



262 ALGEBRA. [§ 474. 

fact that there may be limitations in the problem, expressed or 
implied, which do not appear in the eq^uation. 

Hence, in solving problems, only those values should be 
retained as answers which satisfy all the conditions of the 
given problem. The arithmetical values that will satisfy a 
problem are posUivef and usually the first values found. 

474. In some cases a change in the wording of a given 
problem will form an analogous problem, to which the absolute 
value of the negative root found is an answer. Thus, if Prob- 
lem 3 above be so changed as to state, that, if 4 less oxen had 
been bought, each would have cost $ 5 more, the values of x 
would be 20 and — 16, the latter being inadmissible. 

Imaginary roots indicate that the problem is impossible. 

6. The sum of two numbers is 17, and their product 60. 
What are the numbers? 

7. The sum of two numbers is 72, and their product is 10 
times as great. What are the numbers ? 

8. The sum of two numbers is 50, and the sum of their 
squares is 1282. Find the numbers. 

9. The difference of two numbers is 7, and their product is 
2340. Find the numbers. 

10. The sum of two numbers is m, and their product is n. 
What are the numbers? 

11. The sum of two numbers is m, and the sum of their 
reciprocals is n. What are the numbers? 

12. Find a number such that 5 times its square increased by 
10 times the number itself will equal 495. 

13. Find three numbers such that the second will be one 
half of the first, and the third one third of the first, and the 
sum of their squares will be 441. 



§474.] QUADRATIC EQUATIONS. 263 

14. A certain number of persons pay together a bill of 
$ 190, each paying f 9 less than the number of persons. How 
many were there, and how much did each pay ? 

15. A piece of groimd one rod longer than broad contains 
1190 square rods. What is the length ? 

16. An army corps consisting of 12,850 soldiers was formed 
into two squares, one of which had 10 more men in a side 
than the other. How many men in each square ? 

17. A man made a journey of 48 miles in a certain number 
of hours. If he had traveled 4 miles more per hour, he would 
have made the journey in 6 hours less time. H!ow many miles 
per hour did he travel ? 

18. A farmer bought a number of sheep for $ 80. If he had 
bought 4 more sheep for the same money, he would have paid 
$ 1 less per head. How many sheep did he buy ? 

19. A man, being asked his age, replied, "If to the square 
root of my age you add ^ of my age, the sum will be 26 years." 
What was his age ? 

20. Two couriers, A and B, start at the same time to go to 
a place 90 miles distant. A traveled 1 mile per hour faster 
than B, and reached the place 1 hour before B. At what rate 
did each travel ? 

21. A merchant bought a number of fur robes for $150, 
and then sold them at $ 18 a robe, and thus gained on each 
robe twice the cost of a robe. How many robes did be buy ? 

22. A farmer has two square fields. The side of one is 2^ 
rods longer than the side of the other, and the fields together 
contain 1131^ square rods. How many more square rods in 
the larger field than in the smaller ? 

23. A man sold a farm for f 3150, and afterwards bought 
another farm containing 7 more acres for the same money, at 
$ 5 less per acre. How many acres in the farm sold, and how 
much did he receive per acre ? 



264 ALGEBRA. [§ 474. 

24. A certain number of pieces of cloth cost $ 1260, each 
piece costing $9 more than 5 times the number of pieces. 
How many pieces were there^ and how much did each cost ? 

25. A certain number of persons equally engaged in a 
business transaction lost $96,000; but, four of them becoming 
insolvent, each of the rest had to pay 1(4000 more than his 
fair share. How many persons were engaged in the business ? 

26. A jeweler sold a watch for $ 144, gaining on it as much 
per cent as the watch had cost him. How much did it cost 
him? 

27. A number of horses were bought for $ 1800. Had 3 
more been obtained for the same money, each would have cost 
$ 30 less. How many horses were bought ? 

28. A cistern supplied by two pipes could be filled by one 
alone in 5 hours less than by the other alone, and both 
together could fill it in 6 hours. In how many hours could 
each fill it alone ? 

29. A steamer performed its down trip of 150 miles at a 
certain rate per hour. On the return trip, going 3 miles an 
hour slower, it took 2^ hours longer. What was the rate 
down the river ? 

30. A man bought a certain amount of sugar for $ 66 ; but, 
if sugar were to rise one cent per pound, he would obtain 50 
pounds less for the same money. How much sugar did he buy ? 

31. A drover bought a certain number of sheep for $483. 
Reserving 20 of the number, he sold the rest for $ 432, gaining 

1 on each. How many sheep did he buy ? 



32. What is the price of eggs per dozen when 3 less in 25 
cents' worth raised the price 5 cents per dozen ? 

Find a general formula for the above problem, putting a, 6, 
c, for the above numbers. 



§477.] SIMULTANEOUS QUADRATIC EQUATIONS. 266 



CHAPTER XVI. 

SIBIULTANEOUS QUADRATIC EQUATIONS. 

476. The solution of simultaneous equations of the second 
degree with two unknown numbers involves the elimination of 
one of the unknown numbers, and the forming of an equation 
with but one unknown number. This may lead to an equation 
of a higher degree than the second, usually of the fourth, 
which cannot in general be solved by quadratic methods. 

476. There are, however, several cases in which the solution 
of such simultaneous equations can be effected by equations of 
the second degree. The three cases of most frequent occur- 
rence are 

I. When one of the given equations is simple. . 
II. When the equations are homogeneous and quadratic, 
III. When the equations are symmetrical with respect to the 
unknown numbers, 

A few solutions will sufficiently illustrate the process in each 
case. 

477. I. One of the given equations simple. 

1. Solve the equations \^ + 3xy-f^2% (1) 

^ ( 4.x-y = 7, (2) 

From (2), transposing, y = 4 x — 7 ; (3) 

substituting value of y in (1), x^ + 3«(4 « - 7) - (4 « - 7)2 = 29 ; 
whence 3 x^ - 35 x = - 78. (4) 

Solving (4), x = 8|or3; 

substituting value of x in (2), y = 27| or 6 ; 

.«. X = 8|, y = 27| ; or x = 3, y = 6. 



266 ALGEBRA. [§ 47a 

If preferred, the values of x and y may be braced in corresponding 

naiia thus • / * = ^' ^ = ^» 
pairs, thus. |^^g^^^^27§. 

478. In like manner simultaneous quadratic equations^ when 
one is simple, may be solved by finding in the simple equation 
the value of one of the unknown numbers in terms of the other, 
and substituting in tfie other equation. 

Solve the following groups of equations : 

2 <x» + 3/' = 89, g (5x^-23^ = 93, 
\x-y = 3. ' \3x-~4y = -l. 

3 <x'-f=16, ^ ia^ + 2f = 9, 
' \2x-\ry = 13. ' }.x + 2y = 5. 



\ 



ar^7y2 = l, f5aj«-3a^ + 2^ = 45, 

* x~2y = 2. ' \Sx-^2y = 22, 

6 (a^-2/= = 9, 3 \10x + y=3xy, 

C2a; + 2/ = 14. * \y — x = 2. 



10. 



11. 



7/2 + 20^=11, 

-f 3a; = 9. 



if- 
\2y 

( 5x2 _Sxy + y^- = 2x-Sy-^ 31, 
\5x-2y = ll, 

^^ Ux-2y + (y-6y = 69-2xy, 
(5aj — 4y = l. 

479. II. The equations homogeneous and quadratic. 

An equation is said to be homogeneous when all its terms 
which contain the unknown number are homogeneous (§ 56), 

13. Solve the equations I f + ^ f ^^' (^) 

^ \2xy-f=:3. (2) 

Multiplying the first member of (1) by the second member of (2), and 
the first member of (2) by the second member of (1), we have 

Sx^ + Zxy = 20xy - lOy^ ; (8) 

or 3x2 _ 17 yx = - 10y2. (4j 



§481.] SIMULTANEOUS QUADRATIC EQUATIONS. 267 

Considering 17 y as the coefficient of x in (4), and solving the equation 
as a quadratic, we have 

fl5 = 5yorf y; 

whence 2 acy = 10 y^ or J y^. 

Substituting values of 2icy in (2), 9y2 = 3 ; Jy^ _ ^2 -_ j^a _ 3 . 

whence y = ± iVS; y =±S, 

.-. x=±iVS; x=±2. 

In like manner any two homogeneous equations of the second degree 
may be solved. 

480. Two homogeneous simultaneous equations of the second 
degree may also be solved by assuming y = vx, and substitut- 
ing vx for y in both equations, and then by division or other- 
wise obtaining an equation involving only v. Having found 
the value of v in this equation, the values of x and y may be 
found by substitution. It is, however, believed that the method 
illustrated by Example 13 is simpler. 

Solve the following groups of equations : 

(4 0^-3 0^ = 18. * lxy = 6. 

{2f-xy = 10-4.a^, <2xy -^24. = Sx', 

l2x'-^3xy = 2f. ' (/_a^ = -3. 

<2f^xy = 5, <a^-2xy==9^Sfy 

lx'-2xy = f-\-2. • (3^-40:3^ = 5-63^. 

^^ (a^-23^ = 8, 22 P^-3a:3^ = 4-3^, 

l3f-xy = 4:. ' \a^-2xy = 9-3f. 

18. |a^+2^ = 6i, 23. \<^-y)+y(^-y)=^^s, 

ia^ — xy = 6. ' (7 x(x+y)=72y(x — y). 

481. III. The equations symmetrical. 

An equation is said to be symmetrical with respect to two 
numbers, as x and y, if the numbers can be interchanged 
without destroying the equality. 



16. 



268 ALGEBRA. [§ 482. 

Thus, a? — 2{cy + j^ = 12ia symmetrical ; for, on interchang- 
ing X and y, it becomes jf — 2yx + a^ = 12, a true equation, 
since x^ — 2xy-\-y^ = y^ — 2yx-\-a^ (§ 80). The equation 
0?^ — ^ = 16 is not symmetrical, since aj* — ^ is not equivalent 
to y* — aj*. 

24. Solve the equation \^'^^^7^^' ^^^ 

^ U + y = 12. (2) 

Squaring (2) , x^ +2xy + y^ = 1U; (3) 

subtracting (1) from (3), 2xy = 70; (4) 

subtracting (4) from (1), x^ — 2xy -{-y^ = ^; 

extracting square root, x — y=±2; (6) 

2 X = 14, . •. x = 7 ; 
10, . •• » = 5. 
Substituting 7 for a; in (2), y = 6 ; 

substituting 6 for « in (2), y = 7. 

Hence the values of x and y are x = 7, y = 6; x = 6, y = 7, 



adding (5) and (2), |^^^ 



482. In like manner any two simultaneous symmetrical 
equations of the second degree may be solved by so combining 
them as to obtain the values of the sum and the difference of the 
unknown numbers, 

483. Groups of equations which are symmetrical except in 
the signs of the terms may often be solved by the symmetrical 
method. 

25. Solve the equations < ^ ' )r,i 

(x-y = -'2. (2) 

Squaring (2), x^-2xy+y^ = i] (3) 

subtracting (3) from (1 ) , 2 icy = 48 ; (4) 

adding (1) and (4), ofi -\- 2 xy -\- y^ = 100 ; 

extracting square root, x-{-y = ±10; (6) 

adding (2) and (6), 2 a; = 8 or - 12, .*. a; = 4 or - 6 ; 

subtracting (2) from (6), 2 y = 12 or - 8, /. y = 6 or — 4. 

.*. X = 4, y = 6 ; or ac = — 6, y = — 4. 



§485.] SIMULTANEOUS QUADRATIC EQUATIONS. 269 

26. Solve the equations |^ + 2^ = ^^^ 0) 

(x + y = 5. (2) 

Dividing (1) by (2), x^-xy + y^ = 7; (-3) 

squaring both members of (2), x^ -\- 2 xy -\- y^ = 2b ; (4) 

subtracting (3) from (4), 3 a;y = 18 ; 

whence ojy = 6. (5) 

Subtracting (5) from (3), 7^-2xy + y^=l; 
extracting square root, « — y = ± 1 ; (6) 

adding (2) and (6), 2 a; = 6 or 4, /. ac = 3 or 2. 

Substituting 3 or 2 for a; in (2), yz=z2 or 3. 

484. Groups of symmetrical equations often fall under the 
first case, and are readily solved by substitution. 

Solve the following groups of equations : 
\xy = 35, \x-\-y = 7. 

\xy = — 15. \x — y = 3, 

29. \<^+f = Qh 36. fa^-«^ + 2^=19, 
(^ + y = — 1. ix — y = S, 

30. |«' + 2' = 5, 3g |a^-3/« = 37, 
( a!*y + xjf* = 30. {x — y = l. 

32, (0^ + 2/^ = 29, 33^ j a^ + xy -]-f = 175, 



27. 



28. 



{ 



U-y = 3. la^-2^ = 875. 

SPECIAL METHODS. 

485. The preceding methods of solving simultaneous equa- 
tions of the second degree are called general because they 
apply to classes of equations. There are, however, many 
simultaneous equations not falling under these cases, which 
are readily solved by special artifices. Some equations that 
do fall under these cases may be solved more elegantly by 
special methods. 



270 ALGEBRA. [§ 485. 



1. Solve the equations i ^i /2_aa 



a^ - y* = 369, (1) 

(2) 

Dividing (1) by (2) , x^-y^ = 9; (3) 

adding (2) and (3), 2 x* = 50 ; .-. jb = ± 6. 

Subtracting (3) from (2), 2 y^ = 32 ; .-. y = ± 4. 
Hence a! = 6, y = 4; orx=— 5, y= — 4. 

This solution gives only the finite roots of equations (1) and (2). 

2. Solve the equations |«* + »* = ^'^^ 0-) 

^ I x-\-y = 5. (2) 

Raising (2) to the 4th power, we have 

«* + 4 ie8y + 6 a:ay2 + 4 acy* + y* = 626 ; (3) 
adding (1) and (3), and dividing by 2, 

«* + 2 a% + 3a;V + 2 xy3 + y* = 361 ; 
extracting the square root, x^ + xy + y^ = ± 19. (4) 

We now have two equations, (2) and (4) , which can be readily solved 
by substitution or by the symmetrical method. 



1 



3. Solve the equations 

a2 + 2a^ + i» + 3y = 76, (1) 

f + x^y = 24, (2) 

Adding (1) and (2), x^-\-2xy + y^ + 2x-^ 2y = 99; (3) 

factoring parts and adding 1, (x + y)^ + 2(a: + y) + 1 = 100 ; 
extracting square root, x + y + 1 = ± 10. 

/. X + y = 9 or - 11. (4) 

Equations (2) and (4) can now be solved by substitution, the first 
general method. 

Solve the following groups of equations : 

^ (a^-^f^l6, ^ <a^ + f = m, 

\x + y = S, ' \x + y=2l2, 

(x^-f=.S2, (0^-2^ = 544, 

Xx-y = 2, ' la^ + 2/* = 34. 

(a^^f = 19, (aj*-y*=1280, 

• U-y = l. * la^-f = 32. 



§486.] SIMULTANEOUS QUADRATIC EQUATIONS. 271 

a^4-y* = 706, -^ (aj* + 2/* = 272, 



10. 



11. 



12. 



13. 



faj* + y* = 706, ^^ (aj* + 2/* = 2 
\x-\'y = S. ' \x — y=:2. 

(a^^3xy + f = B9, (a^ + y* = 2657, 

W + 2^ = 29. * U + y = ll. 

|a^ + l = a? + y, ^g (aj« + 2ajy + y + 3aj = 73, 

j«* + y* + « + y = 18, j^^ <oi^ + xy = a^ + ab, 
\xy = 6. ' \ j^ + yx = b^ -\- (ib. 



PROBLEMS INVOLVING SIMULTANEOUS QUADRATICS. 

486. The statement of problems involving two or more un- 
known numbers has been sufficiently illustrated under the head 
of simple equations. Several of the problems below can be 
solved by the use of only one unknown number, but their 
solution is facilitated by the use of two. 

1. The difference of two numbers is 7, and the difference 
of their squares is 119. What are the numbers? 

2. The sum of two numbers is 18, and the sum of their 
squares is 170. What are the numbers? 

3. The sum of two numbers is 25, and the difference of 
their squares is 175. What are the numbers? 

4. Find two numbers such that the first increased by twice 
the second is 24, and the sum of their squares is 149. 

5. The sum of the squares of two numbers exceeds twice 
their product by 9, and the difference of their squares is 1 
less than their product. Find the numbers. 

6. The sum of six times the greater of two numbers and 
five times the less is 50, and their product is 20. Find the 
numbers. 

7. The product of two numbers diminished by their sum 
is 17, and the sum of their squares is 65. Find the numbers. 



272 ALGEBRA. [§ 48«. 

8. Find two numbers such that their sum is 19^ and the 
sum of their cubes 1843. 

9. Find two numbers such that their difference is 4, and 
the difference of their cubes 988. 

10. The product of two numbers multiplied by their sum 
is 180, and the sum of their cubes is 189. Find the numbers. 

11. If a certain number expressed by two digits be multi- 
plied by the sum of its digits, the product will be 160 ; and, if 
the number be divided by four times its unit digit, the quo- 
tient will be 4. Find the number. 

12. What number divided by the product of its two digits 
is 5^, but, when 9 is subtracted from it, the resulting number 
is expressed by the two digits in an inverse order ? 

13. The area of a rectangular field is 300 square rods, and 
the length of its diagonal is 25 rods. Find the length of the 
sides. 

14. The sum of the diagonal and the longer side of a 
rectangle is three times the length of the shorter side, and 
the difference in the lengths of the two sides is 4 yards. What 
is the area of the rectangle ? 

15. The area of the floor of a certain hall is 5375 sq. ft., and 
its length is 4 feet less than three times its breadth. What 
are the dimensions of the floor ? 

16. A certain number of sheep were bought for $ 468 ; but, 
after 8 of them had been reserved, th« rest were sold at an 
advance of f 1 a head, and $ 12 were gained on the lot. How 
many sheep were bought ? 

17. A vessel can be filled in 6 hours by two pipes running 
at the same time, but one pipe can fill it alone in 5 hours less 
than the other. How many hours does each pipe require to 
fill it? 



§ 491.] INEQUALITIES. 273 



CHAPTER XVII. 
DTEQUALITIES. 

487. The expression a>h denotes that a is greater than 6, 
and a<b denotes that a is less than b (§ 39). The sign > or < 
is called the sign of inequality. 

488. An inequality is an expression consisting of two unequal 
numbers connected by the sign of inequality. Thus 4 > 3 and 
a? < y are inequalities. 

489. Two inequalities are said to subsist in the same sense 
when their first members are both greater or both less than 
their second members. Thus, a > 6 and c > d subsist in the 
same sense. 

490. Two inequalities are said to subsist in a contrary sense 
when the first member is the greater in one, and the less in the 
other. Thus, a > 6 and c < d subsist in a contrary sense. 

Inequalities are also called inequations; and two inequalities which 
subsist in the same sense are also said to be of the same direction, since 
the signs point in the same direction ; and two inequalities that subsist in 
a contrary sense are also said to be the reverse, since the signs point 
iu opposite directions. 

It is assumed in this chapter, unless the contrary be stated, that the 
letters denote real and positive numbers. 

491. 5 > 3, and 5 -|- 2 > 3 -h 2 ; and, generally, if a > 6, then 
a-\-c>h-{-c. Likewise 5 > 3, and 5 — 2 > 3 — 2 ; and, gener- 
ally, if a > 6, then a — c'>h — c. Hence, if the same positive 
number be added to or subtracted from both membei^s of an in" 
equality^ the resulting inequality will subsist in the same sense. 

WHJTJ£*S JLLQ, 18 



274 ALGEBRA. [§ 492. 

492. It follows that a term can be transposed from one 
member of an inequality to the other, as in an equation, pro- 
vided its sign he changed, 

498. If a>6, then 2a>26, 3a>36, and ac>6c; and, if 

a > 6, then - > -, - > -, and - > — Hence, if both members of 
2233 c c 

an ineqvxdity be multiplied or divided by the same positive num^ 
ber, the resulting inequality will subsist in the same sense, 

494. A positive number is greater, algebraically considered, 
than any negative number ; and, of two unequal negative num- 
bers, the less numerically considered is the greater algebraically 
(§ 67). Thus, 2 >- 7, and - 2 > - 7. 

Hence, if both members of an inequality be multiplied or 
divided by the same negative number, the resulting inequality will 
subsist in a contrary sense. Thus, 2<,5, but 2x(— 3)>5x 
(-3); also -2>-5, but -2 x(-3)<-5 x(-3). 

495. It follows, that, if the signs of both members of an in- 
equality be changed^ the resulting inequality wiU subsist in a 
contrary sense. 

496. If a>by c>dy and e >f then a-\-c-\-e>b-\-d +/, 
and axcxe^bxdxf Hence, if two or more inequalities 
that subsist in the same sense be added member to member , or 
multiplied member by member, the resulting inequalUy wHl subsist 
in the same sense, 

497. If an inequality be subtracted from, or divided by, 
another inequality in the same sense, the resulting inequality 
may or may not subsist in the same sense. Thus, 5 <S, and 
1 < 5, but 5 - 1 > 8 - 5; 4 < 6, and 1 < 3, but 4 -f- 1 > 6 ^ 3. 

These operations are to be avoided when the sense of the 
resulting inequality cannot be determined. 

498. 'if a > b, then a^ > b\ a^ > l^, and a" > b\ Hence, if 
both members of an inequality of positive numbers be raised to the 
same power, the resulting inequality will subsist in the same sense. 



§ 501.] INEQUALITIES. 276 

499. If the members of an inequality are negative^ the 
same odd powers will subsist in the same sense, and the same 

 even powers in a contrary sense. Thus, if — a > — 6, then 
(__ of > (- 6)^ but (- of < (- h)\ 

500. If a* > 6*, then a^ > W and a > 6. Hence, if the same 
root of both members of an inequality of positive nurabers be taken, 
the resulting inequality will subsist in the same sense. 

601. An inequality, like an equation, may be simplified by 
certain transformations ; and an inequality is said to be solved 
^when a limit to the value of the unknown number is found. 

1 . Simplify the inequality ^ - 2 > 3 + ^. 

Transposing - 2 and ^, ^ - ^>5; 

clearing of fractions, and uniting, 5 a; > 30 ; 

dividing by 6, oc > 6. 

Hence 6 is a limit of the value of x. 

2. Simplify 2a! + |-4>^-|. 



3. Find the limits of x, when given < 



-!+»<!+„ (1) 



Transposing in (1), ^ - -< a - 6 ; 

b a 

clearing of fractions, <mc — 6x < a^ft — a6^ ; 

factoring, (a — h)x <(o — h)ah ; 

dividing by a — 6, x < a6. 

Clearing (2) of fractions, ox — 6a; > o* — o5 ; 

factoring and dividing by a — 6, x > a. 

Hence the limits of x are a and ah. 

4. Find the limit of aj in — < "" % 

b X b» 



276 ALGEBRA. [§ 502. 

Find the limits of a; in 



5. 



(10x<3x + 49. 



X a — b 



8. < 



6. < 



ic-h5>| + 55. 



2 3 ^2"~ ^^2"" 



•1 



a — 6 a: 

ax — ox<i . 

X 

aa -\- bx > ab + h^, 



fAx-2 2-Ax 



"^ X . ^x 



7. < 



3 



a 



3x-2<^ + ^. ^*' )? + ?>« + '» 



2 o 6 a 



\i 



502. TA6 »t^m q/* the squares of any two unequal numbers 
is greater than twice the product of the numbers. 

For let a and b be any two unequal numbers. Then, since 
(a — by is positive whether a> or < 6, we have 

(a-by>0. 
Expanding, a* — 2a6-|-6*>0; 

transposing —2ab, a^ + 6^ > 2 ab. 

11. Show that the sum of any fraction whose terms are 
unequal and its reciprocal is greater than 2. 

Let - be any fraction in which a > or < 6. Then, by § 502, we may 
b 

assume that o^ + 6^ > 2 ab. 

Dividing by a6, ^ + ->2. 

b a 

Assume that a and b are positive unequal numbers, and 
show that 

12. a^b + ab^>2aV. ,. a±b^ 2ab 

14. — — — ^ 



13. a« + 6«>a26 + a6l 2 a + 6 



§ 506.] RATIO. 277 



CHAPTER XVIII. 

RATIO, PROPORTION, VARIATION. 

RATIO. 

503. Ratio is the relation of one number to another of the 
same kind expressed by their quotient. Thus^ the ratio of 

a to 6 is -• 



Every fraction expresses the ratio of its numerator to its denominator ; 
and every integer expresses the ratio of itself to unity. 

604. A ratio may be expressed by writing a colon (:) be- 
tween its two terms. Thus, the ratio of a to 6 is expressed 
by a : 6, read " the ratio of a to 6," or, briefly, " a to 6." 

The ratio of a to 6 is expressed by a : 6, and the ratio of 6 to a by 6 : o ; 
but the ratio between a and 6 or 6 and a is expressed hy a:b or bict. 
Thus, the ratio of 3 to 6 is f, and the ratio of 6 to 3 is } ; but the ratio 
between 3 and &^r 6 and 3 is | or f . 

505. The first term of a ratio is called the antecedent; and 
the second term, the consequent. The two terms of a ratio 
taken together are called a couplet. 

The antecedent of a ratio is the dividend ; and the consequent, the 
divisor. 

506. When the antecedent equals the consequent, as a: a, 
the ratio equals unity, and is called the ratio of equality. 

When the antecedent is greater than the consequent, as 8 : 5, 
the ratio is greater than unity, and is called a ratio of greater 
inequality. 



278 ALGEBRA. [§ 507. 

When the antecedent is less than the consequent, as 5:8, 
the ratio is less than unity, and is called a ratio of less 
inequality. 

When the antecedent and the consequent are interchanged, 
the resulting ratio is the inverse of the given ratio. Thus, 
6 : a is the inverse of a : 6. 

607. Since 2 = 5L2i^, and 2 = ?L±«, hoth terms of a ratio 
b b xn b 6-5-n 

may be multiplied or divided by the same number without aUer- 
ing the value of the ratio. 

506. If - = r, " ^ ^ = m, and — ?L.= rii; hence multiply- 
b b b -s-n 

ing the antecedent or dividing the consequent of a ratio by a 
number multiplies the ratio by that number. 

509. If - = r, ^^^ or —55— = '^ ; hence dividing the ante- 

b b b xn n 

cedent or multiplying the consequent of a ratio by a number 
divides the ratio by that number. 

510. Eatios may be compared by reducing the fractions that 
express them to a common denominatory and comparing the result- 
ing numerators. 

a o 

Thus, to compare a : b with c:d, we reduce - and - to 

ad be b d 

— and — respectively, and then decide that a:b > or = 
bd bd 

or < c : d, according as ad > or = or < be. 

511. If a>b, ^L±^<«; and if a<b, ^>?. 

b -{-n b b -{-n b 

For, reducing the ratios to a common denominator, we have 
a-^n _ ab-{-bn •, a _ db -\- an 
b + n^bip + ny b~ b(b-\-n) 

If a>6, (ab + bn)<i(ab + an), and hence 5L±_5<?; but, if 

b -\- n b 

a<b, (ab -h bn)>(ab -{■ an), and hence ^"^^ >~. 

6 -f- n b 



§ 515.] RATIO. 279 

Hence a ratio of greater inequality is diminished, and a ratio 
of less inequality increased, by addiiig the same number to both 
terms of the raJbio, 

512. If a > 6, ; > -• ; and, if a < ft, ; < -• 

b —n b b — n b 

For, reducing the ratios to a common denominator, we have 

a — n _ ab — bn -j a _ ab — an 
b — n b(b — n) b b(b — n) 

If a>b, (ab — bn) > (ab — an), and hence "~ > - ; but, if 

a<b, (ab — bn) < (ab — an), and hence ~" < -• 

b — n b 

Hence a ratio of greater inequality is increased, and a ratio 
of less inequality diminished, by subtracting the same nuraber 
from both its terms. 

The principles stated in §§511, 512, may be thus illustrated : 

(1) |±|<4 ^^d ^>?; (2) ^-^>K and ?-:i2<?. 
^^3 + 2 3 4 + 2 4 ^^3-2 3 4-2 4 

613. Generally, since every ratio may be expressed as a 
fraction, whatever operations on the terms of a fraction affect its 
value, will in like manner affect the value of the corresponding 
ratio, 

514. A compound ratio is the product of two or more ratios. 
Thus, the ratio axiibd is the product of a : 6 and c : d, and is 
called compound. 

Two or more ratios may be compounded by taking the product 
of the fractions that express them. For example, the ratios 
oi a:b, c:d, and e :f, are compounded by taking the product of 



-, -, and —• Thus, r X - X — = 



ace 



b' d' f ' b d f bdf 

615. A ratio, as a : 6, may be compounded with itself, a^ : &' 



280 ALGEBRA. [§ 516. 

is called the dujf^iccUe of the ratio a : 6 ; cfib^, the triplicate of 
aib'j and so on. 

The ratio Va : -y/b or a^ : b^ is called the subduplicate of 
a:b; and v^a : ^6 or a^ : 6*, the subtriplicate of a : 6. 

516. When one or both terms of a ratio are incommensurable 
(§ 371), the ratio is said to be iticommensurable. Thus, 1 : -V2 
is an incommensurable ratio. 

Problbhs. 

1. What is the ratio of 2 lb. to 2 oz. ? Of f 0.75 to $ 3 ? 

2. Arrange in descending order of magnitude 

4:5, 7:3, 12 : 4, 3:8, 5 : 12, 7:5. 

3. What is the ratio compounded of 2 : 3 and 15 : 16 ? Of 
7 : 6 and 24 : 35 ? 

4. Two numbers are in the ratio of 3 to 5; but, if each 
be increased by unity, their ratio becomes 11 : 18. Find the 
numbers. 

5. Two numbers are in the ratio of 8 to 7; but, if each 
be increased by 6, their ratio becomes 6 : 5. Find the num- 
bers. 

6. Give the duplicate and triplicate ratios of 5 : 7, the 
subduplicate ratio of 81 : 25, and the subtriplicate ratio of 
729 : 343. 

7. Reduce the following ratios to their lowest terms : 

63:45, 138:124, a": ax, rrv" - vJ" : m^ + n\ 

8. Find two numbers in the ratio of 4 to 5, such that their 
difference is to the difference of their squares as 1 to 27. 

9. Find two numbers such that the ratio of their sum to 
the sum of their squares will be as 11 to 195, the ratio of the 
numbers themselves being 4 : 7. 

10 o Find X so that the ratio of a? to 1 may be the duplicate 
of the ratio of 8 to x. 



§ 521.] PROPORTION. 281 



PROPORTION. 

617. A proportion is the expressed equality of two ratios. 

This equality may be expressed by the sign : : or the sign =. 
Thus, the equality of the two ratios a : b and c : d may be 
expressed by a:b::c:d or by a:b = c:d, each being read 
" the ratio of a to 6 equals the ratio of c to d," or, briefly, " a is 
to 6 as c is to d" 

518. Since each ratio has two terms (an antecedent and 
a consequent), a proportion necessarily consists of four terms, 
the first and third being antecedents, and the second and 
fourth, consequents. 

The first and fourth terms of a proportion are called the 
extremes; and the second and third, the means. Thus, in 
a:b::c:dy a and d are the extremes ; and b and c, the means. 

619. Four numbers are said to be proportional, or in pro- 
portion, when the ratio of the first to the second equals the 
ratio of the third to the fourth. Thus, a, b, c, d, are propor- 
tional when a:b = c: d. The four terms of a proportion are 
called proportionals, the fourth term being called the fourth 
proportional. 

620. Numbers are in continued proportion when the ratios 
of the first to the second, the second to the third, the third 
to the fourth, etc., are equal. Thus, a, b, c, d, e, etc., are in con- 
tinued proportion when a:b = b :c = c:d = d:e, etc. 

621. Since the ratio a : b may be expressed by the fraction 

-, and c\d by -, the proportion a\b = c:d is identical with 
b d 

the equation - = -; and, by simple transformations of this 

b d 

fundamental equation, the following propositions in proportion 
are proved. 



282 ALGEBRA. [§ 522. 

Propositions. 

522. If four numbers are in proportion, the product of the 
extretnes equals the product of the means. 



Let 


a:b = c:d'j 


then 


a_c 
b~d 


Multiplying by 6d, 


cul = bc. 



523. It follows from the above, that, if any three terms of a 
proportion are giveiiy the other term can be found. 

For, if ad=bcy then a = — , d = — : b = — , c = — : and 

da c b 

hence either extreme of a proportion may be found by dividing 

the product of the means by the other eoctreme; and either mean 

may be found by dividing the product of the extremes by the otiier 

m^an. 

Find the value of x in the following proportions : 



1. 


5:8 


: : 15 : 0?. 


6. 


x:10+x::6:9. 


2. 


x:5 , 


: 6 : 10. 


6. 


aj : 15 - a? : : 30 : 15. 


3. 


5:x\ 


: 7 : 10. 


7. 


6 : « : : 24 : aj + 6. 


4. 


4:6 


: : a? : 4. 


8. 


4 : 18 : : a - 3 : a? + 4. 



5S4. If the product of two numbers is equal to the product of 
two other numbers, the four nunibers are proportionals ; and any 
twi) of them may be made the extremes, and the other two the 
means, of a proportion. 

Let ad = 6c. 

Dividing by M, od^bc 

^ ^ ' bd bd' 

whence r = -; i.e., a:6 = c:<t 

d 



c 


'd' 


b__ 


d. 


a 


c ' 


b_ 
d 


c ' 



§ 527.] PROPORTION. 288 

The equation ad = &c, if divided successively by cd, ac, and 

dc, gives 

a b ' y jt 

- = - ; i.e., a : c = : a ; 

i.e., b:a = d:c; 

— = - ; i.e., b: d=:a:c. 
d c 

Let the pupil obtain these proportions from ad = bc, and 
compare them with the next two propositions. 

Let the pupil illustrate the above, and also each of the fol- 
lowing eight propositions, by means of numbers. 

625. If four numbers are in proportion, they will be in propor- 
tion taken alternately ; i.e., the first term will be to the third as the 
second term to the fourth. 

Let a\b = c\d\ 

then _ = _. 

b d 

Multiplying by -, "" ~ 3 » ^®*' ^ • ^ = & : d. 

C C Cv 

526. If four numbers are in proportion, they will be in propor- 
tion taken inversely; i.e,, the second term wiU be to the first as the 
fourth term to the third. 

Let a:b = c:d; 

then 2 = i. 

b d 

Dividing 1 by each member, 1 -i- - = 1 -5- - ; 

b d 

whence - = - > !•©•> b:a = d:c. 

a c 

627. When a -h 6 is to 6 as c + d is to d, the numbers a, b, c, 
and d are said to be in proportion by composition. 



284 ALGEBRA. [§ 52a 

When a-^b is tobssc — d is to dy the numbers (hb,Cy and d 
are said to be in proportion by division. 

When a-^-b istoa — 6 as c + distoc — d, the numbers 
a, by Cf and d are said to be in proportion by composition and 
division. 

628. If four numbers are in proportion^ they are in proportion 
by composition or division. 

Let a'.b = c:d\ 

then . 55 = f. 

b d 

Adding ± 1 to each member, 7 ± 1 = - ± 1 ; 

b d 

whence a±b^c_±d 

b d ' 

that is, a±b\b = c±d'.d. 

529. If four numbers are in proportion^ they are in proportion 
by composition and division. 

Let a:b = c:d\ 

then (§ 528) ^^ = ^±^, (1) 

b d 

and 2l=±^^.^=1. (2) 

b d ^ ^ 

Dividing (1) by (2), member by member, 

— 6 c — d^ 
that is, a-f^-a — & = c4-d:c — d. 

530. In a series of equal ratios, the sum of the antecedents is tc 
the sum of the consequents as any antecedent is to its consequent. 

then, by § 522, ab = ba, ad = be. af= be, ah = bg. 

Adding and factoring, a{b -\- d +/4- h^= b(a + c + e 4- g') ; 
that is, a-hc-f^-f5r:6 + d +/+ * = a : 6. 



or 



§ 534.] PROPORTION. 285 

631. The pivduct of the corresponding terms of two or more 
proportions are in proportion. 

Let a:b = c:dy 

and e:f=g:hy 

a c ji e g 

b d f h 

Multiplying member by member, tt = „ > 

of dh 

that is, ae:bf=cg: dh. 

532. If a, b, c, d, are in continued proportion (§ 620), a is to 
c as a* 18 to b*, and a is to d cw a^ is to h\ 

Let a:b = b:c = c:d, 

a b c 
6 c d' 

whence 2x- = -x-; 

b c b b 

a a * 9 1.Q 

or -=-^; i-e.; a:c = a^:Cr, 

c W 

b c d b b b 

CL fit 

or - = -i: ; i.e., a:d = a^:b^. 

d b^' ' 

533. When three numbers, a, 6, c, are in continued propor- 
tion, b is called a mean proportional to a and c, and c a third 
proportional to a and 5. 

534. The mean proportional to two numbers is equal to the 
square root of their product. 

Let a : 6 = 6 : c ; 

then r = -' 

b c 

Clearing of fractions, b^ = a>c; 
whence b = Vac. 



286 ALGEBRA. . [§535. 

535. Several of the foregoing theorems are useful in the solu- 
tion of certain numerical problems, and also certain equations. 

536. Prove the following propositions, and illustrate each 
with numbers : 

I. If two proportions have the same couplet in each, the 
other couplets will form a proportion. 

II. If two proportions have the same antecedents, the con- 
sequents are in proportion. 

III. If three numbers are in proportion, the ratio of the first 
to the third is the duplicate ratio of the first to the second. 

IV. If the first two terms of a proportion be multiplied by 
m, and the last two terms by w, the resulting products will 
be in proportion. 

Find the value of a: in the following proportions : 

1. a-f 1: a; + 6 = 0? + 17: aj-h 19. 

2. 3a + 3:3aj-4 = 6aj + l:5aj-a 

3. x-\-a:x + b = x + c:x-\'d. 

4. mx + a : gaj + 6 = mx -{- c:qx + d. 

6. (aj+7)(aj-4) : (a;+3)(aj-l) = (a;+l)(a;-4) : (a?+l)(a;-5). 

Problbms. 

1. Find a fourth proportional to 4, 6, 12. 

2. Find a fourth proportional to ^, ^, ^. 

3. Find a mean proportional to 4 and 9; 4 and 16. 

4. Find a third proportional to 9 and 12; 7 and 14. 
6. Find a mean proportional to | and ^; f and 30. 

6. Find a third proportional to a^ and 2<ibi ocy and 3 ajy*. 

7. Divide the number 20 into two parts such that the ratio 
of their squares will be as 9 to 4. 

8. Divide the number a into two parts such that the ratio 
of their squares will be as m^ to n*. 



§538.] VARIATION. 287 

9. Divide the number 32 into two parts such that the 
quotient of the greater divided by the less will be to that of 
the less divided by the greater in the ratio of 25 to 9. 

10. Find two numbers such that the sum of their squares 
will be to the difference of their squares as 17 to 8, and the 
difference of their squares to the difference of their cubes as 

8 to 49. 

11. Divide $ 121 among A, B, and C, so that A's share will 
be to B's as 4 to 5, and B's to C's as 9 to 8. 

12. The area of a rectangular field is 3 acres, and its length 
to its breadth as 6 to 5. Find the dimensions. 

13. The number of dollars A has is to the number B has as 

9 to 6. By obtaining one half of A's money, B will have $ 2 
more than A had at first. How much money has each ? 

14. In a square-hewn block of stone containing 5 cubic 
feet, the length is to the breadth as 9 to 5, and the breadth to 
the thickness as 5 to 3. Find the dimensions of the stone. 

15. A man's age is to that of his wife as 9 to 8. Ten years 
ago their ages were as 13 to 11. What are their ages ? 

16. Of two houses, one cost f 1000 more than the other, and 
the ratio of their prices was as 3 to 2. Find the cost of each. 

VARIATION. 

537. When two quantities are so related that one increases 
or diminishes in the same ratio as the other, the first is said to 
vary as the second. 

Thus, if a train of cars run a miles per hour, the distance 
run in 2 hours will be 2 a miles, in 3 hours 3 a miles, and so 
on ; that is, the distance will vary as the time varies. 

538. This variation of two numbers is denoted by the 
sign oc written between them, called the sign of variation, and 
read " varies as." Thus, a oc 6 is read " a varies as 6." 



288 ALGEBRA. [§ 639. 

539. A number which in jiny particular problem cjianges its 
value is called a yariable, and a number that has a fixed value 
is called a constant. Thus, in the example given above, the 
number of miles per hour (a) is a constant, while the numbers 
denoting time and distance are both variables. 

540. When two variables are so related that if one be given 
the other can be found, one is said to be a function of the 
other. Thus, for example, if a steamer sails at a given speed, 
the distance sailed in a certain time will depend on the time, 
and, if the time be given, the distance can be found; and 
hence the distance is a function of the time. 

541 . When one number varies as another, their correspond- 
ing values have a constant ratio. 

Thus, if tti, Og, «3, etc., denote the increase of one variable, 
and 6i, 62? ^s? ^tc, the corresponding increase of the other, so 
that 

^ = |i, (l);and 5^=^«, (2) 

then, from (1), ?Li = ^f? . ^nd from (2), -^ = ?i. (§ 525.) 

bi 62 ^2 ^8 

Hence -i = -? = -^ . . . = a constant ratio. 

h bi bs 

Hence, if aocft, their common ratio is found by dividing 
a by bf or a2 by 62, or Og by b^ and so on. If m denotes this 

common ratio, 7 = m, and a = bm, 

b 

542. There are many ways in which two variables may be 
related. Four of the more important cases are here presented. 

543. I. When two numbers so vary that their corresponding 
values have a constant ratio, they are said to vary directly. 

Thus, if a Qc & directly, ^ = wi, their constant ratio, and 

a = bm. Hence, if, in - = m or a= bm, a = 12 and 6 = 4, 
m = 3 and a = 3b. 



§ 545.] VARIATION. 289 

1. A workman earns $40 a month. Upw much will he 
earn in 5 months? In 12 months? 

Which number in this problem is the constant ? Give the 
two variables. Which is the function of the other ? 

2. The base of a rectangle is 10 inches. What will be its 
area if the altitude be 1 inch ? 3 inches ? 8 inches ? 

Which number is the constant? Give the two variables. 
Which is the function of the other? 

544. II. When one number varies as the reciprocal of 
another, the numbers are said to vary inversely. Thus, when a 

varies inversely as 6, aoc - ; and then a = ~ x m, and ab = m. 

For example, the time required to do a given work varies 
inversely as the number of workmen employed. If 2 men can 
do the work in 6 days, 4 men can do it in 3 days ; that is, 
twice as many men will do it in one half of the time. 

3. Three men can dig a ditch in 12 days. How long will it 
take 6 men to dig it ? 9 men ? 12 men ? 

Which number in this problem is the constant ? What num- 
bers are the variables ? Which is the function of the other ? 

4. If a quantity of oats will feed 4 horses 9 days, how many 
horses will it feed 3 days ? 

545. In some cases one quantity varies inversely as the 
square of another. Thus, aoc— • 

For example, the illumination of a candle decreases inversely 
as the square of the distance from it. If it gives a certain 
illumination at a distance of 1 foot, the illumination at 2 feet 
will be only \ as much ; at 3 feet, only ^ as much ; and so on. 

6. If the illumination of a gas jet at 26 feet is a?, what will 
it be at 60 feet ? At 100 feet ? 
white's alo. — 19 



290 ALGEBRA. [§ 546. 

6. If the attraction of a magnet for a piece of iron at the 
distance of ^ of an inch is x, what will be the attraction at -^-^ 
of an inch ? ^ of an inch ? ^ of an inch ? 

546. III. When one number varies as the product of two 
other numbers, it is said to vary as the two others jointly. 
Thus, if accbc, a varies jointly as 6 x c. 

For example, the area of a rectangle varies as the product of 
its base and altitude. 

7. If the area of a rectangle with a given base and altitude 
is X, what will be its area when the base and altitude are each 
doubled ? When the base is trebled and the altitude doubled ? 

The pressure of gas varies directly as its density, and also as 
its temperature ; and hence, if the pressure of gas at a given 
density and temperature is represented by x, its pressure when 
its density is doubled, and its temperature increased one half, 
will be expressed hyxx2x^ = Sx. Hence, 

8. If the pressure of gas at a given density and temperature 
is 15 pounds to the square inch, what will be its pressure if the 
density be trebled, and the temperature reduced one half ? 

547. IV. A number is said to vary directly as a second num- 
her, and inversely as a third, when it varies as the product of 
the second and the reciprocal of the third. 

Thus, a varies directly as 6, and inversely as c, when 

a oc 6 X - ; that is, when a : 6 x - is constant, 
c c 

When a number varies as the quotient of two nambers, it varies 

directly as the dividend, and inversely as the divisor. Thus, if ax-, a 
varies directly as &, and inversely as c. ^ 

9. The volume of gas varies as the absolute temperature, and 
inversely as the pressure. If the volume is represented by x 
when the pressure is 15 and the temperature 300, what will be 
the volume when the pressure is 20 and the temperature 350 ? 



§ 550.] VARIATION. 291 

648. In all four of the cases of variation presented above, 
the constant can be determined when any one set of corre- 
sponding values is given. 

Thus (1), if aQC&,- = w; (2), if aoc-, ab = m; (3), if accbc, 

a J /*\ 'i? r 1 h dc 

— = m : and (4), if a oc o x -, a = - x m. .-. — = m. 
be c c b 

549. If a depends only on b and c, and aoc 6 when c is 
constant, and aocc when b is constant, then, when both b and c 
vary, a oc be. 

Let a,b,c; a', b\ c, a", b', c', — be three sets of correspond- 
ing values. 

Then, since c is in the first and second, — - = - • (1) 

a' b' 

also, since 6' is in the second and third, -- = -• (2) 

a" c' 

Multiplying (1) by (2), 5x^ = ^.5 

^ a be . a a^^ 

whence — = — • .*. — = — • 

a" 6'c' be b'e' 

Hence a varies as be. 

For example, the area of a rectangle varies as its altitude 
when its base is constant, and as its base when its altitude is 
constant, and as the product of its base and altitude when both 
vary. 

650. The simplest method of solving problems in variation 
is to convert the variations into equations. 

For example, if a oc 6 and 6 oc c, show that a oc c. 

By § 543 we have a = bm (1), and b = cn (2), m and n being 
constant ratios. 

Multiplying (1) by (2), ab = bcmn ; 

whence a = cmn, .*. aocc. 



292 ALGEBRA. [§ 550. 

10. If XQcy, and x = 5 when y = S, find x when y = 9. 

11. If a<x h, and & is 9 when a is 6, what is h when a is 9 ? 

12. a oc 6 and 6 x c : show that ococbK 

13. a oc - and bcc-i show that a oc c 

6 c 

14. xcKyz: ii x = 2 when 2/ = 4 and 2 = 3, what will a equal 
when y — 2 and z = 9? 

15. If the area of a rectangle is x when its base is a and 
its altitude by what will be its area when its base is 3 a and its 
altitude 1 6 ? 

16. The volume of a sphere varies as the cube of its diameter. 
If the volume of a sphere 2 inches in diameter is 4.188 cu. in., 
what is the volume of a sphere 5 inches in diameter ? 

17. The area of a sphere varies as the square of its diameter, 
and the surface of a sphere 5 inches in diameter is 78.54 sq. in. 
What is the surface of a sphere 10 inches in diameter ? 

18. If the volume of a sphere varies as the cube of its diam- 
eter, how many spheres 3 inches in diameter equal a sphere 12 
inches in diameter ? 

19. The area of a circle varies as the square of its diameter. 
How many circles 4 inches in diameter equal one 20 inches in 
diameter ? 

20. The velocity of a falling body varies as the time during 
which it has fallen from rest. If the velocity of a falling ball 
at the end of 2 seconds is 64 feet, what will be its velocity at 
the end of 6 seconds ? 

21. The distance a body falls from rest varies as the square 
of the time it falls. ' If a ball falls 144 feet in 3 seconds, how 
far will it fall in 12 seconds ? 

22. The quantity of water that flows through a circular pipe 
varies as the square of the pipe's diameter. If 10 gallons a 
minute flow through an inch pipe, how many gallons per 
minute will flow through a 4-inch pipe ? 



4 56«i.j PKOGRESSIONS. 293 



CHAPTER XIX. 

PROGRESSIONS. 

551. A series is a succession of numbers formed according 
to some fixed law, called the law of the series. The successive 
numbers are called the terms. of the series. 

552. A series that consists of a limited number of terms is 
called a finite series, and one that consists of an unlimited 
number of terms is called an infinite series. If a finite series be 
considered as continued indefinitely in either or both direc- 
tions, it becomes an infinite series. 

ARITHMETICAL PROGRESSION. 

553. An arithmetical progression is a series in which each 
term is obtained by adding a constant number to the preced- 
ing term. The constant number added is called the common 
diiference. An arithmetical progression is denoted by A. P. 

554. If the common difference is positive, the series is said 
to be incjreasing. Thus the series 3, 5, 7, 9, 11, ••• is an increasing 
arithmetical progression, in which the common difference is 2. 

555. If the common difference is negative, the series is said 
to be decreasing. Thus the series 12, 9, G, 3, 0, — 3, — 6, •••, is 
a decreasing arithmetical progression, in which the common 
difference is — 3. 

556. The first and last terms of a finite arithmetical series 
are called the extremes; and the intermediate terms, the 



294 ALGEBRA. [§557. 

arithmetical means. Thus, in the series 2, 6, 10, 14, 18, with 
live terms, 2 and 18 are the extremes ; and 6, 10, and 14, the 
arithmetical means. 

557. If a denotes the first term of an arithmetical progres- 
sion, and d the common difference', then a -f d will denote the 
second term, a-\-2d the third term, a-\-Sd the fourth term ; 
and so on to the nth term, which will be denoted by a-|-(w— l)d. 

If the series consists of n terms, and the nth or last term be 
denoted by Z, we have the equation 

l=a-{'(n-- l)d (A) 

558. This formula contains four symbols, denoting as many 
different numbers. If any three of these numbers are given, 
the fourth may be found by suhstitviing the given numbers for 
their symbols in {A), and solving the resulting equations, 

1. rind the thirteenth term of the A. P. 6, 10, 14, 18, .... 

Here n = 13, a = 6, and <f = 10 - 6 = 4. 
Substituting 13 for n, 6 for a, and 4 for d, in (^), we obtain 

Z = 6 + (13 - 1) 4 = 6 + 12 X 4 = 54. 

Hence the thirteenth or last term is 54. 

The common difference in a given arithmetical series may evidently 
be found by subtracting any term from the term which next follows it. 

2. In an A. P. of 30 terms, the last term is 8, and the com- 
mon difference is 2. Find the first term. 

Substituting 30 for n, 8 for Z, and 2 for d, in (^), we obtain 

8 = a + (30 - 1) X 2. 
.-. a = 8 -58 =-50. 

3. Find the number of terms of an A. P. whose first and 
last terms are 5 and 65 respectively, and common difference 3. 

Substituting 5, 65, and 3 for a, I, and d respectively, in {A), we obtain 

65 = 5 + (n - 1) 3 ; 
whence n — 1 = 20. .*. n = 21. 



§ 558.] PROGRESSIONS. 295 

4. Find the common difference when the first and last terms 
of an A. P. of 64 terms are 9 and 16 respectively. 

Substituting 64, 9, and 16 for n, a, and I respectively, in (^), we obtain 

16 = 9+ (64-l)d. 

, 16-9 1 

•*• " = — :n — = :;;• 
63 9 

5. Insert 10 arithmetical means between 2 and 35. 

Tlie first step is to find the common difference, and hence the problem 
is similar to the above. Since there are 10 mean terms, the whole number 
of terms is 12. 

Substituting 12, 2, and 35 for n, a, and I respectively, in (^), we obtain 

35 = 2 + (12-l)d. 
.-. d = }f = 3. 
Hence the required series is 2, 6, 8, 11, 14, 17, 20, 23, 26, 29, 32. 35. 

6. The sixth term of an A. P. is 2, and the thirteenth term 
is 23. What is the first term ? 

Make the given sixth term the first term of a series of 8 terms in vv^hich 
a = 2, 2 = 23, and n = 8. 

Substituting 2, 23, and 8 for a, I, and n respectively, in {A), we obtain 

23 = 2+ (8-l)d = 2 + 7(Z. 
.'. d = S. 

Now consider the sixth term (2) the last term of a series of 6 terms, 

and we have 

2 = a + 5 X 3. 

.*. a = — 13, the first term of the series. 

7. Find the fifteenth term of the A. P. 17, 14, 11, .... 

8. Find the sixteenth term of the A. P. — 12, — 7, — 2, «... 

9. Find the fifteenth term of the A. P. 7|, 6|, 6^, .... 

10. Given w = 15, d = 4, 1 = 70: find a. 

11. Given n = 50, c? = f , ? = 5| : find a. 

12. Given n = 105, d = .07, I = - 2.36 : find a. 



296 ALGEBRA. [§ 559. 

13. Given cl = 3, a = 12, / = 72: find n. 

14. Given d = - J, a = -5J, ? = -101||: find*. 

15. Given d = .5871, a = .29, i = 59 : find n. 

16. Given 11 = 22, tt = 4, /=130: find d. 

17. Given n = 58, a = .33, / = 75: find d. 

18. Insert 7 arithmetical means between 5 and 29. 

19. Insert 8 arithmetical means between 1 and — 5. 

20. Insert 9 arithmetical means between — 6 and 0. 

559. A general formula for finding the sum of n terms of an 
arithmetical progression may be obtained as follows: 

Let s denote the sum of the terms, and I the last term ; then 

« = a + (a + d)-h(a + 2d) + (a + 3d)-|-.-. -f-^ (1) 
also s = ;-f (Z-d)4-(i-2d)-f (Z-3d)-|-... +0, (2) 

the sura in (2) being written in reverse order. 
Adding (2) to (1), term to term, we obtain 

25= (a4-0 + («+0 + («+0+(«+0H h(aH-0 ton terms. 

Hence 2s = n(a-^l), 

and « = ^(a + r). (B) 

560. This formula also contains four symbols, and may be 
employed to obtain solutions for as many classes of problems. 

21. Find the sum %i 15 terms of an A. P. that has for its 
extremes 7 and 45. 

Here n = 15, a = 7, Z = 45; and substituting these values for n, a, 
and I respectively, in (i?), we obtain 

« = \- (7 + 45) = 15 X 26 = 390. 

it 



§ 562.] PROGRESSIONS. 297 

22. The sum of 40 terms of an A. P. is — 660, and the last 
term is — 36. Find the first term. 

Substituting 40, — 560, and — 36, for n^ s, and {, in (jB), we obtain 

-660 = — (a- 36), 

whence ^^720-560^3^ 

20 
The last term of an A. P. is found in like manner when s, a, and n are 
given. 

23. The sum of the terms of a series is 272, and the ex- 
tremes are — 13 and 45. Find the number of terms. 

Substituting — 13, 46, and 272, for a, I, and s respectively, in (5), we 
obtain 

272 = I (-13 + 46), 

1 272 ,^ 

whence n = — = 17. 

16 

24. Given a = 11, 1 = 25, n = 14: finds. 

25. Given a = -13, Z = 105, n = 63: finds. 

26. Given a = — 3.7, I = — 1.3, w = 42 : find s. 

27. Given s = 598, Z = 39, w = 26 : find a. 

28. Given s = 1295, Z = 80, n = 37 : find a. 

29. Given s = 1184, a = 11, n = 32 : find Z. 

30. Given s = 116.375, a = - 3.5, w = 19 : find Z. 

31. Given s = 336, a = 3, 1 = 25: find w. 

32. Given s = 30.225, a = .75, Z = 1.2 : find w. 

33. Given s = 147 V5, a = 3 V5, Z = 18V5: find w. 

561. The fundamental formulas (A) and (B) contain, in all, 
five symbols; and such is the relation between the numbers 
represented, that, if any three of them are given, the other two 
can be found. 

562. Of the twenty classes of problems thus arising, eight 
can be solved by a single formula, as above exemplified; but 
the remaining twelve classes require both formulas, since they 



298 ALGEBRA. [§ 562. 

each involve the numbers denoted by d and », given or required, 
and d occurs only in {A)y and 8 only in (B). 

34. Find the sum of 30 terms of the series 7, 10, 13, •••. 

Here n = 30, a = 7, and d = 3, and s is required. 

Substituting their values for n, a, and d, in {A) and (£), we obtain 

I = 7 + (30 - 1) X 3 = 04, (1) 

8 = V (7 4- = 105 + 151. (2) 

Substituting in (2) the value of Z in (1), we have 

s = 105 + 15 X 94 = 1515. 



f 



35. How many successive terms of the series 11, 9, 7, 
must be taken, that their sum may be 27 ? 

Here a = ll,d = — 2, and s = 27, and n is required. 

Substituting their values for a, d, and s, in {A) and (B), we obtain 

1 = 11 + (»-l)xC-2)=13-2n, (1) 

27 = -(ll + 0- .-. 54 = lln+Z». (2) 

Substituting in (2) the value of I found from (1), simplifying, etc., we 
obtain n2-12n = -27; 

solving the equation, n = 9 or 3. 

For n = 9, the A. P. is 11, 9, 7, 5, 3, 1,-1, — 3, — 5, the sum being 
27. 

For n = 3, the A. P. is 11, 9, 7, the sum being 27. 
To find the value of I, substitute both values of n in (1), obtaining, for 
n = 9, Z = 13 - 2 X 9 = - 5 ; for n = 3, Z = 13 - 2 x 3 = 7. 

PROBLEMS. 

Find the sum of 

1. 30 terms of the A. P. 5, 7, 9, .... 

2. 16 terms of the A. P. 9, 6, 1, .... 

3. 36 terms of the A. P. 1^, 2, 2^, .... 

4. 15 terms of the A. P. — 2J, ~1|, —1, .... 

5. 19 terms of the A. P. 5.3, 6.1, 6.9, .... . 

6. 100 terms of the A. P. 1, 2, 3, .... 

7. n terms of the A. P, 1, 2, 3, •••, 



§ 562,] PROGRESSIONS. 299 

8. Find the sum of n terms of the A. P. 1, 3, 5, •••. 

9. Find the nth term of each series in 7 and 8. 

10. The extremes of an A. P. are 2 and 30, and the common 
difference is 2. Find the sum of the series. 

11. The first term of an A. P. is 10, the number of terms 10, 
and the sum of the terms S5, Find the common difference. 

12. The common difference of an A. P. is 2, the last term 
23, and the number of terms 11. Find the sum of the series. 

13. The common difference of an A. P. is 2, the number of 
terms 12, and the sum of the terms 90. Find the last term. 

14. The number of terms of an A. P. is 13, the common dif- 
ference — 7, and the sum of the terms 39. Find the first term. 

15. The last term of an A. P. is 52, the common difference 
5, and the sum of the series 297. Find the number of terms. 

16. The fourth term of an A. P. is — 1, and the tenth term 
is 3. What is the first term ? What is the sum of the series ? 

17. A man paid $3000 in 6 installments. The first was 
$ 400, and the last $ 600, all the payments being in an 
A. P. What was the amount of each intermediate payment ? 

18. The cost of sinking a well was $ 45, f 1 being paid for 
sinking the first yard of depth, $ 1.50 for the second, $ 2 for 
the third, and so on. What was the depth of the well ? 

19. A man began saving by putting by 1 cent on New Year's 
Day, 2 cents on the next day, 3 cents on the next, and so on. 
In how many days would he have put by f 98.70 ? 

20. A load of 100 fence posts was laid down at the corner 
of a field. If, leaving 1 post at the corner, a man should 
carry the rest one by one, to lay them down in a straight line 
at intervals of 3 yards, how far would he walk to accomplish 
his task ? 



800 ALGEBRA. £§ 563. 

GEOMETRICAL PROGRESSION. 

563. A geometrical pnogression is a series in which each term 
bears a constant ratio to the preceding one. A geometrical 
progression is denoted by G. P. 

564. This constant or common ratio is called the ratio of the 
aeries ; and the series is increasing or decreasing according as 
its ratio is numerically greater or less than unity. 

Thus, the series 2, 6, 18, •••, is an increasing geometrical 
progression in which the ratio is 3; and 64, 16, 4, •••, is a 
decreasing progression in which the ratio is \. 

565. If the ratio be positive, all the terms will be positive; 
but if the ratio be negative, the terms will be alternately posi- 
tive and negative, as in the series 5, — 10, + 20, — 40, • • •, in 
which the ratio is — 2. 

A geometrical progression is a continued proportion, as defined in § 520. 

566. When a geometrical progression consists of a definite 
number of terms, the first and last are called eztrQmes ; and the 
intermediate terms are the geometrical means. 

A geometrical scries may be considered as extending indefinitely in 
either direction, and it is then called an infinite geometrical series, 

567. If a denotes the first term of a G. P., and r the ratio, 
then ar will denote the second term, aii^ the third term, ai^ 
the fourth term; and so on to the nth term, which will be 
denoted by ar^'^, it being the (n — l)th term after the first. 

Hence the last term (/) in a series of n terms will be desig- 
nated by the formula 

I = ar^-\ {A) 

568. This formula, like the corresponding one in arith- 
metical progression, contains four symbols, denoting as many 



§ 568.] PROGRESSIONS. 801 

numbers ; and, if any three of these numbers are given, the 
fourth can be found hy substituting the given numbers for their 
symbols in (A), and solving the resulting equation. 

1. Find the ninth term of the G. P. 243, 81, 27, .... 

Here a = 243, n = 9, and r = ^\^ = J. 

Substituting their values for a, n, r, in (^), we obtain 

i = 243 X f i V= 243 X = — • 

\SJ 243 X 27 27 

It is evident that the ratio in a given G. P. is found by dividing any 
given term by the one preceding it. 

2. Find the first term of the G. P. whose eleventh term is 
3072, and whose ratio is — 2. 

Substituting their values for n, ?, r, in (A), we obtain 

3072 = a X (- 2)10 _ « x 1024 ; 
whence a = 3072 -f- 1024 = 3. 

3. Find the ratio of the G. P. whose first and eighth terms 
are 1715 and yttt respectively. 

Substituting their values for a, I, w, in (^), we obtain 

^^^=1715r7; 
6 1 



whence r = 



2401 X 1715 2401 x 343 



=^^ 



1 



2401 X 343 7 

4. Insert 4 geometrical means between — -^ and 144. 

The first step is to find the ratio, as in the preceding problem. The 
number of terms is 4 + 2, or 6. 

Substituting their values for a, 2, w, in (A), we obtain 

144 = -^Vx^, 
whence . r = — 6. 

Hence the required means are 

A^ -ih W» - 4F; or J, -}, 4, -24. 

5. Find the eighth term of the G. P. 1, 2, 4, •••. 

6. Find the tenth term of the G. P. 28, 7, J, .... 



S02 ALGEBRA. [§ 56a 

7. Find the seventh term of the G. P. 1, — ^, t^, •••. 

8. Given I = 1458, w = 6, r = 3 : find a. 

9. Given I = 960, n = 7, r = — 2: find a. 

10. Given Z = — 567, n = 5, a = — 7 : find r. 

11. Given Z = |f|, n = 7, a = 2: find r. 

12. Insert 3 geometrical means between 2 and ff. 

13. Insert 7 geometrical means between ^ and 9. 

14. Insert 5 geometrical means between ^ and -j-TT-g^. 

15. Show that the geometrical mean between two numbers 
is the square root of their product. 

669. When a, I, and r are given in order to find n, the pro- 
cess leads to an exponential equation; that is, an equation in 
which the unknown number occurs as an eocponent. The solu- 
tion of this form of an equation can usually be effected only 
by the use of logarithms, as shown in § 613. 

570. A general formula for finding the sum of n terms of a 
G. P. may be obtained as follows : 
Let s denote the sum of the terms ; 

then s = a -h ar -h ar* 4- ar^ -|- ••• + ar'^'K (1) 

Multiplying by r, rs = ar-\- ar^ -|- ar^ + '•• -f aY^~^ + ai^. (2) 

Subtracting (1) from (2), 

r8 — 8 = ait^ — a, or s (r — 1) = a (r* — 1) ; 

«^ 1 

whence s = a x (B) 

r — 1 

671. This formula also contains four symbols, and may be 
employed for the solution of as many classes of problems ; but 
in two cases it may lead to equations of a higher degree than 
the second. 



§ 573.] PROGRESSIONS. 303 

16. Find the sum of 8 terms of the G. P. 2, 6, 18 -.. 
Here a = 2, n = 8, r = 3. 
Substituting their values for a, n, r, in (B), we obtain 

s = 2 X ^^^^ = 38 - 1 = 6560. 
3-1 

17. The sum of 10 terms of a G. P. is — 1705, and the ratio 
is — 2. Find the first term. 

Substituting their values for n, r, 8,iD. (B), we obtain 

-1706 = ax^^^^ — ^ = ax -341; 

— ^ — 1 

whence a = — = 6. 

-341 

18. Find the sum of 6 terms of a G. P. the first and last 
terms of which are 3 and -^ respectively. 

Substituting their values for n, a, I, in (^), we obtain 

r = i. 
Substituting their values for a, n, r, in (B), we obtain 

« = 3 X W = 5|l- 

672. When r < 1 numerically, the formula (B) is more con- 
venient in the form 

1 — r» 

5 = a X 

1 — r 

When r = l, 8 = a(l + r + r^ + r8-f... + r»-^), and hence s = an. 

Infinite Series. 

573. If r < 1, the value of r" decreases as n increases ; and 
hence, as n is indefinitely increased, r** is indefinitely dimin- 
ished, and thus approaches zero for its limit Therefore the 
above formula becomes a 

—^ — being the limit towards which s approaches when r < 1, 
1 — r 

and both a and r are positive, and n is indefinitely increased. 

This limit is properly called the limit of the sum of a 



804 ALGEBRA. [§ 573. 

converging geometrical series as n approaches infinity, but for 
convenience it is usually called the limit of the series to itifinity. 
An infinite number is denoted by the symbol oo. 

For converging and diverging series, see Chapter XXI., p. 336, note. 

19. Find the limit of the sum of 1 + ^ -f ^ ••• to infinity. 

Here a = 1, r = J ; and hence, by (B'), 8= =2. 

1 — i 

This may be illustrated by taking a slip of paper, say 2 inches long, 
dividing it into two equal parts, taking away one of these parts, and 
bisecting tlie other ; and so on indefinitely. It is evident that the sum of 
the parts taken away can never exceed 2 inches. 

20. Find the sum of 10 terms of the G. P. 2 - 6 + 18 . 

21. Find the sum of 7 terms of the G. P. 75 + 15 -h 3 + .-% 

22. Find the sum of 8 terms of the G. P. 1 — | + f .. 

23. Given w= 8, r= 2, s = 765: find a. 

24. Given n= 7, r=— 3, « = 547 : find a. 

25. Given n = 10, r = !> * "^ ^^ff • ^^^ ^• 

26. Given n = 7, r = .5, s = 1943.1 : find a. 

Find the limit of the sum, to infinity, of 

27. 1 + ^ + ^+.... 30. 3 + f-f^+-. 

28. 1+1 + 1+.... 3,^ l+l + i. .... 

29. f + | + A+-. ^ ^ 

32. Find the value of the recurring decimal .53434 + •... 

The repeating part, .03434 +, is an infinite geometrical series, the first 
term being ^Ujj^ ^*^® second rjjjjjy^, and so on. 

Substituting ^ooir for a, and yj^ for r, in {B'), we obtain 

Hence the value of the decimal is /^^ + ^^q = UJ. 

A recurring decimal is also called a circulating or a repeating decimal. 
The repeating part, called the repetend, is denoted by a dot placed over 
the first and the last of its figures ; thus, .534. 



§ 575.] PROGRESSIONS. 305 

Find the value of 

33. .24i. 35. .354. 37. 8.90l. 

34. 6.53. 36. .45i24. 38. 54.321. (See also ^ 6SS.) 

574. Of the twenty possible cases in geometrical progres- 
sion, eight can be solved by either (A) or (5), as shown above. 
The twelve remaining cases — that is, all those cases in which 
both the numbers denoted by I and s are involved — require the 
use of both formulas. 

39. The extreme terms of a G. P. are 3 and 192, and the 
ratio 2. Find the sum of the terms. 

Substituting their values for a, Z, and r, in (A) and (5), we obtain 

192 = 3 X 2»-i. .-. 2"-i = 64, or 2« = 128. (1) 

s = 3 X ^*~ ^ ' .'. substituting 128 for 2», s = 381. (2) 

40. Find the seventh term of the G. P. whose ratio is f , and 
the sum of the series 42||. 

Substituting their values for n, r, «, in (-4) and (B), we obtain 

i = ax(i)6 = axW» (1) 

2059 W/-1 2059 . ,^ 

__=ax-I|^ = ax-^. .-. a = t. (2) 

Substituting J for a in (1), 

575. When the number denoted by n is required, the process 
leads to an exponential equation (§ 569) ; and in some cases, if 
n > 2, to equations of higher degree than the second. The 
foregoing examples may sufficiently indicate how to proceed in 
soluble cases. Sometimes the value of n in an exponential 
equation is readily found by inspection, as in 2" = 128, occur- 
ring in the solution of Example 39 above. 

WHITR*S ALO. 20 



306 ALGEBRA. [§ 675. 

PROBLEMS. 

1. Given a = 4, 1 = 2916, n = 7: find s and r. 

2. Given a = 5, 1 = 320, n = 7 : find s and r. 

3. Given 7' = 4, Z = 1024, 7i = 9: find s and a. 

4. Given r = 2, Z = 20480, n = 14; find a and s. 

5. Given r = 3, s = 2391484|, a = ^: find Z. 

6. Given i = 65536, s = 74898^, a = ^: find r and w. 

7. What is the sum of the series 2, ^, f, •••, to infinity ? 

11 1 

8. What is the sum of 1, — — , — , — -^> •••> to infinity ? 

9. The extremes of a G. P. are 2 and ^^, and the ratio 
is ^. What is the sum of the series ? 

10. The sum of the third and fourth terms of a G. P. is 18, 
and the difference of the third and fifth terms is — 36. Find 
the series. 

11. A population increases yearly in geometrical progression, 
and in 4 years is raised from 10,000 to 14,641. Find the ratio 
of annual increase. 

12. There are 3 numbers in geometrical progression whose 
sum is 62 ; and the sum of the first and second is to the sum 
of the second and third as 1 to 5. Find the numbers. 

13. The difference between the first and second of 4 num- 
bers in geometrical progression is 15, and the difference 
between the third and fourth is 540. Find the numbers. 

14. A person wishes to send f 9950, besides as much more 
as will cover the express charges for the whole sum at the rate 
of ^ %. How much should he send in all ? 

15. Some grains of wheat found in the crop of a wild fowl 
being planted, one germinated and produced 50 sound grains. 
These again being sown produced a crop of 2500 grains. Of 
how many grains would the crop of the sixth year consist, sup- 
posing the grain to increase every year at the same rate ? 



§ 578.] PROGRESSIONS. 807 

16. An elastic ball falls from a sufficient height to rebound 
30 feet, and at each successive rebound rises ^ of the distance 
of the previous one. How many feet will the ball pass over in 
5 rebounds ? How many feet before it comes to rest ? 

17. There are 4 numbers in geometrical progression, and 
the first is 21 less than the fourth, and the difference of the 
extremes divided by the difference of the means is 3^. Find 
the numbers. 

HARMONIC PROGRESSION. 

676. Three numbers are said to be in harmonic proportion 
when the first is to the third as the difference of the first and 
second is to the difference of the second and third. 

Thus, if a, h, c, are in harmonic proportion, 

a\c = a — b\h — c, or a\c=h — a\c — h. 

yt*t. An harmonic progression is a series in which the first 
of any three consecutive terms is to the third as the difference 
of the first and second is to the difference of the second and 
third. An harmonic progression is denoted by H. P. 

Thus, if a, h, c, d, e, are in harmonic progression, 

a:c=:a—b:h—c\ b:d=b—c:c—d; and c: e=c— d: d— e. 

578. If a, b, c, are in harmonic progression, we have, by 

definition, 

a:c = a — 6:6 — c; 

whence c(a — b) = a(b — c). 

a — b b — c 



Dividing by abc, 



whence 



ab be * 

1111 
b a c b 



Hence, if given numbers are in harmonic progression^ their 
reciprocals are in arithmetical progression. 



308 ALGEBRA. [§ 579. 

579. Problems relating to harmonic series are usually best 
solved by writing the reciprocals of the terms as an arithmetical 
progression, and then solving the resulting equations. 

1. The second term of an H. P. is 2, and the fourth term 6. 
Pind the series. 

The second aud fourth terms of the corresponding A. P. are } and (. 
Let a be the first term, and d the common difference. Then 

o + d = i, 

whence o = J, and d = — \. 

The A. P. is }, J, J, i, etc.; 

hence the H. P. is f , 2, 3, 6, etc. 

2. Insert 3 harmonic means between 3 and 16. 

The first and fifth terms of the corresponding A. P. are |^ and ^ ; and 
hence d = (^^ - J) -^ 4 = - ^J^. 

Thus the A. P. is J, A, ft» A» A- 

Hence the 3 harmonic means are -^^, 5, ^. 

3. The first term of an H. P. is 2, and the third term is 6. 
What is the second term ? 

4. The second term of an H. P. is &, and the third term is c. 
What is the first term ? 

6. The first term of an H. P. is 2, and the fourth term is 6. 
Find the mean terms. 

6. Insert 5 harmonic means between \ and -j^. 

7. Insert 4 harmonic means between f and f. 

8. Insert 4 harmonic means between \ and -j^. 

9. The first term of an H. P. is 1, and the third term ^. 
Pind the sixth term. 

10. -, -, -, are in arithmetical progression. Show that 
a h c 

a — b :b — c = a:a 



§ 582.] LOGARITHMS. 809 



CHAPTER XX. 
LOGARITHMS. 

580. The logarithm of a number is the exponent of the 
power to which a fixed number, called the base, must be raised 
in order to produce the given number. 

Thus, since 3* = 81, 4 is the logarithm of 81 to the base 3; 
since 8* = 512, 3 is the logarithm of 512 to the base 8 ; and, 
generally, if a* = m, x is the logarithm of m to the base a, 

581. The base of the system of logarithms in common use is 
10, the basis of the decimal notation ; and the system is called 
the common system. 

682. Since 10^ = 1, 10 - ^ = 1 = .1, 

' 10 ' 

10^ = 10, 10-2 = -L=.oi, 

10^ =100, 10 -3 = -L = .001, 

> 103 ^ 

10« = 1000, 10-* = -i-= .0001, 

' 10* ' 

10* = 10000, and so on, 

the numbers 0, 1, 2, 3, •••, are the logarithms of the successive 
positive integral powers of 10 ; and — 1, — 2, — 3, — 4, •••, are 
the logarithms of the successive negative integral powers of 10. 
Thus (log being an abbreviation for logarithm), 

log 1 = 0, log.l =-1, 
log 10 = 1, log .01 =-2, 
log 100 = 2, log .001 = - 3, and so on. 



810 ALGEBRA. [§ 583. 

583. The logarithms of numbers between the integral 
powers of 10 are evidently fractional. Thus, the logarithms 
of the numbers between 

100 and 1000 are 2 -h a decimal, 

10 and 100 are 1 -h a decimal, 

1 and 10 are -f a decimal, 

1 and .1 are — 1 + a decimal, 

.1 and .01 are — 2 -h a decimal, 

.01 and .001 are — 3 -f a decimal, 

and so on. 

Thus, with — above negative characteristics, the logarithm of 

5 is 0.69897, .5 is 1.69897, 

25 is 1.39794, .025 is 2.39794, 

225 is 2.35218, .00225 is 3.35218. 

684. It is evident from the above that a logarithm consists 
of two parts : (1) a positive or negative integral number, called 
the characteristic ; and (2) a positive fractional part, called the 
mantissa. Thus, in the logarithm 2.78176 (log 605), the 2 is 
the characteristic, and the decimal .78176 is the mantissa. 

The characteristic is so called because it shows, as will be explained 
later, the number of orders, integral or decimal, in the corresponding 
number. 

685. It is also evident that the characteristic of the log- 
arithm of a number greater than 1 is positive^ and that the 
characteristic of the logarithm of a number less than 1 and 
greater than is negative. The mantissas of all logarithms 
are positive. 

586. There are three ways of writing a logarithm when its 
characteristic is negative, as follows : 

(1) 2.41162; (2) .41162-2; (3) 8.41162-10. 

12.33244 ; .33244 - 12 ; 8.33244 - 20. 



§ 590.] LOGARITHMS. 311 

The first method is simple, and is used herein. The sign — 
is written over the characteristic to show that it alone is nega- 
tive, the mantissa being always positive. 

587. The number that corresponds to a given logarithm is 
called the antilogarithm, which is abbreviated as antilog. 
Thus, 605 is the antilog of the logarithm 2.78176. 

588. The logarithms of a series of numbers arranged in 
tabular form is called a table of logarithms (§ 611). 

The logarithms of numbers, as now found arranged in tables, were 
originally calculated by very laborious methods, consisting essentially in 
repeated extractions of the square root. These methods now possess only 
an historic interest, since methods have been devised by which logarithms 
can be calculated with great facility. These methods, however, depend 
upon principles the proof of which belongs to a more advanced treatise 
on algebra. 

PRINCIPLES. 

589. The use of logarithms as a means of facilitating certain 
numerical computations involves principles which we now pro- 
ceed to establish; and since these principles are the same, 
whatever may be the base of the system, let us denote the 
base by the general symbol a. 

690. Let m and n denote any two numbers, and x and y 
their respective logarithms to the base a; then, by definition, 

a' = m. (1) 

a' = n, (2) 

Multiplying together (1) and (2) member by member, we have 

a*+y = mn ; 
whence x + y = log mn, by definition. 

Hence the logarithm of a product is equal to the sum of the 
logarithms of the factors. 

It follows that the logarithm of a composite number is the 
sum of the logarithms of its factors. 



312 ALGEBRA. [§ 591. 

691. Dividing (1) by (2) member by member, we have 

a"" = — ; 
n 

whence x — y = log— 

n 

Hence the logarithm of a quotient is equal to the difference of 
the logarithms of dividend and divisor. 

It follows that the logarithm of a common fraction is the 
logarithm of its numerator minus the logarithm of its denomi- 
nator. 

The sums and differences of numbers cannot be found by logarithms. 

592. Raising both members of (1) to the power denoted by 

J), we have 

a*" = m'; 

whence px = logm^. 

Hence the logarithm of a power is equal to the logarithm of 
the bajie of that power multiplied by the exponent of the power. 

593. Extracting the rth root of both members of (1), we have 

a'^ = S/m ; 

whence - = log S/m. 

r 

Hence the logarithm of a root is equal to the logarithm of the 
power divided by the index of the root, 

594. If in (1) we make m = 1, the corresponding value of x 
will be zero, since a° = 1, whatever a may be (§ 123). 

Hence, in aU systems of logarithms^ logl = 0, 

595. If, again, in (1) we make x = l, then, since a^ = a, or 
log a = 1, it follows that in all systems of logarithms the log- 
arithm of the base is unity. 



§ 599.] LOGARITHMS. 813 

596. The application of the foregoing principles to numerical 
computations by logarithms involves (1) the obtaining of the 
logarithms of numbers from the tables, and (2), when a loga- 
rithm is given, the finding of the corresponding number or 
antilog. These processes will now be explained. 

ARRANGEMENT AND USE OF TABLES. 

597. The simplest method of presenting a table of logarithms 
would be to arrange the numbers in their natural order in ver- 
tical columns, and place opposite each number its logarithm; 
but tables thus arranged would be altogether too voluminous 
for use. Various expedients, therefore, are employed in order 
to save space in tables, and also time in their use. The great 
advantage, indeed, of adopting 10 as the base of the common 
system arises from the fact that tables of logarithms to that 
base can be presented in a very compact form. 

698. It has already been shown that the logarithm of any 
number between two integral powers of 10 lies between the 
integers which are the logarithms of those powers (§ 583). 
Thus, the logarithm of the number 3265, which is between 10^ 
and 10*, must lie between 3 and 4, the exponents of the given 
powers of 10. This logarithm is, in fact, 3.51388. 

It is thus seen that the positive characteristic of a logarithm 
is always one less than the number of integral orders in its 
antilog. 

699. The mantissas of a series of logarithms increase as 
their antilogs increase from one power of 10 to another. This 
increase of mantissas is shown below for the antilogs from 1 
to 10^ and from 10^ to 10^, only three figures of each mantissa 
being given. 

..rl 2 3'4 5 6 7 8 9 10 
Antilog I ^^ 2^ ^^ ^^ ^^ g^ ^^ g^ ^^ ^^^ 

Mantissa .000 .301 .477 .602 .698 .778 .845 .903 .954 .000 



814 ALGEBRA. £§ 60a 

600. The numbers 326500, 32650, 3265, 326.5, 32.65, 3.265, 
.3265, .03265, etc., expressed by the same sequence of figures, 
but differing in the position of the decimal point, may all be 
derived from any one of the set by multiplying or dividing by 
some power of 10. Hence the logarithms of any two of these 
numbers can differ from each other only by tlie logarithm of 
some power of 10; that is, by some integer, positive or nega- 
tive, the mantissa remaining the same for all. Hence 

Numbers expressed by the same sequence of significant figures 
have the same mantissa in their logarithms. 

Thus, log 326500 = log (3265 x lO^) = log 3265 -f log 10* 

= 3.51388 + 2 = 5.51388. 

log .03265 = log (3265 -?- l(f)_ = log 3265 - log 10* 
= 3.51388 - 5 = 2.51388. 

601. Since the increase or decrease of the characteristic by 
so many units manifestly corresponds to the shifting of the 
decimal point so many places to the right or left in the anti- 
log, what has been shown above in regard to one series of 
antilogs and their logarithms will evidently hold good in 
regard to any other similarly related series. 

602. It follows that the characteristic of the logarithm of any 
number is equal to the number of places by which the left-hand 
digit of tJiat number is distant from the units place; and that 
it is positive when the digit lies to the left, iiegative when it 
lies to the right, and zero when it is in the units place. 

Thus, (1) log 3625 = 3.51388, (5) log .3625 = 1.51388, 

(2) log 362.5 = 2.51388, (6) log .03625 = 2.51388, 

(3) log 36.25 = 1.51388, (7) log .003625 = 3.51388, 

(4) log 3.625 = 0.51388, and so on. 

603. It may be adopted as a good rule, that a positive char- 
acteristic is numerically less by 1 than the number of figures 
before the decimal point of the given number; and that a 



§ 605.] LOGARITHMS. 815 

negative characteristic is numerically greater by 1 than the 
number of zeros after the decimal point and before the first 
significant figure. Thus, the characteristic of log 73582 is 4 ; 
the characteristic of log 4.965 is 0; and the characteristic of 

log .00053 is 4. 

As the characteristic can thus always be supplied by in- 
spection of the given number, it is accordingly omitted from 
the tables, which might more appropriately be called tables 
of mantissas than tables of logarithms. 

Write down the characteristics of the logs of the following 
numbers, placing the minus sign over negative characteristics : 

1. 7; 70; 70000; .7; .00007. 3. 5360; .536; 5.36; .00536. 

2. 23; 2.3; .23; 230; .023. 4. 43892; 43.892; 43892000. 

604. The mantissas in the tables herein given (pp. 319-321) 
may be regarded as those of the logarithms of all numbers from 
1 to 1000. The mantissa of log 365, for example, is found in 
the line opposite 36, the first two figures of the antilog, and 
in the column headed by 5, the third figure. The mantissa 
thus found is .56229. 

In the line opposite 36, in the columns headed by 0, 1, 2, ... 9, we find 

, the mantissas .55630, .55751, .55871, .•• .56703, belong respectively to the 

logs of 360, 361, 362, ... 369. In the line opposite 40, the mantissa in 

the column headed is not only the mantissa of log 400, but also of 

log 40, log 4, log .04, etc. 

605. To find the logarithm of a number expressed by not 
more than three figures. 

To the proper characteristic annex the mantissa found opposite 
the first twofi^gures in the column headed by the third. 
Thus, 

(1) log 300 = 2.47712; log 30 = 1.47712; log 3 = 0.47712 ; 

(2) log .03 = 2.47712 ; log 57 = 1.75587 ; log 579= 2.76268 ; 

(3) log 4.78 =0.67943; log 10.5=1.02119; log 2390 =3.37840. 

Note. Always write down the characteristic before seeking the 
mantissa. 



316 ALGEBRA. [§ 606. 

Find the logarithms of the following numbers: 
6. 2, 3, 4, .5, .6, .007, 80, 900, 90000. 

6. 23, 340, 4500, 5.3, .67, .072, .0085. 

7. 121, 306, 272000, 35.4, 4.62, .537, .00643. 

8. .876, 7.65, 65.4, 543, 4320, 32100. 

606. The method of finding the logarithms of numbers ex- 
pressed by more than three figures is made evident by a single 
illustration. Since the number 362.45, which is greater than 
362, is less than 363, the difference between log 362 and log 363 
is greater than the difference between log 362 and 362.45. Let 
d denote the difference between log 362 and log 363; then 
log 362.45 will be log 362 increased by less than d. 

Assuming that the logarithmic difference for a fraction of a 
unit will be the same fractional part of the difference for a 
whole unit, — an assumption generally true as far as two places 
of decimals, — we can find log 362.45 by multiplying d by .45, 
and adding the product to log 362. But log 362, as found in 
the tables, is 2.55871, and the difference between, log 362 and 
log 363 is .00120, and hence log 362.45 is obtained as follows : 

log 362 = 2.55871 
d X .45 = .00120 X .45 = 54 

whence log 362.45 = 2.55925 

Since the mantissa for log 36245 is the same as that for 
362.45 (§ 600), we may find log 36245 by considering 45 a 
decimal, and proceeding in exactly the same way, except that 
we make the characteristic 4 instead of 2. Hence, 

607. To find the logarithm of a number expressed by more 
than three figures, 

To the logaritJim of the number expressed by the first three 
figures, as found in tlie table, add the product of the corresponding 



§ 608.] LOGARITHMS. 317 

tabular difference multiplied by the number expressed by the 
remaining figures regarded as a decimal. 

In the tables of logarithms given below, the difference between each 
mantissa and the next higher mantissa is printed over the first. These 
differences are called tabular differences, 

9. ¥ind log 4.7389. 

log4.73 = 0.67486 d = 92 

d X .89 = .00092 X .89 = 82 JB9 

whence log 4. 7389 = 0.67568 828 

736 
d' = 81.88 

For convenience the tabular difference may be regarded as an integral 
number, and the product be formed as at the right above. We reject the 
88, and since the first rejected figure is not less than 5, we consider the 
81 as 82. Generally increase the fifth figure by 1 when the sixth figure of 
a mantissa is more than 5, and drop the sixth figure when it is less than 6. 

10. Find log .00124. 

log .00124 = 3.09342 

d X .6 = 349 X .6 209 

whence log .00124 = 3.09551 

Find the logarithms of the following numbers : 

11. 96520; 8736; 764.5; 65.48; 5.321; .004512. 

12. 35962; 2045.3; 156.78; 93.125; 8.6545. 

13. .004567; .0056789; .06789; .78901. 

14. .89012; 90.123; 123.45; 2345.6; 34567. 

608. The antilog of a given logarithm is found by practi- 
cally reversing the above process. For example, the antilog of 
4.81491 is obtained (1) by finding the mantissa in the tables, 
and writing the corresponding antilog. The mantissa .81491 
is found in line 65, column 3, and hence the antilog of said 
mantissa is 653. But (2) since the characteristic of the given 
logarithm is 4, there must be five (4 + 1) integral orders in the 



818 



ALGEBRA. 



[§609. 



antilog, and hence the antilog sought is 65300. If the given 
characteristic had been 2, the antilog would have been 653; 
and, if the characteristic had been 0, the antilog would have 
been 6.53 ; and so on. 

609. To obtain the antilog of 2.89387, we find in the tables 
the next lower mantissa, which is .89376, in line 78, column 3. 
We write 783 for the first three figures of the antilog. The 
remaining figures are found as below : 

Given mantissa, .89387 

Next lower mantissa, .89376 

Dividing by d or 55, 55)11.00(.20 

Annexing the quotient .20 to 783, and pointing so as to have 
three (2 -|- 1) integral orders, we obtain 783.2 as the required 
antilog. If the characteristic had been 2, we should have 
placed one cipher (— 2-|-l) after the decimal point, thus obtain- 
ing .07832 as the antilog. 

610. To find the antilog of a given logarithm, 

I. If the given mantissa is found in the tables, write down 
the figures of the corresponding antilog, and determine the posi- 
tion of the decimal point by the given characteristic, 

II. If the exact given mantissa is not found in the tables, sub- 
tract from it the next loiver mantissa, and divide the difference by 
the tabular difference of this lower mantissa. Annex the first two 
figures of the quotient to the antilog of the lower mantissa, and 
place the decimal point as determined by the given characteristic. 

Find the antilogs of the following logarithms : 



15. 0.30103 

16. 1.47712 

17. 1.07918 

18. 2.13113 

19. 3.64098 



; 0.47712; 


, 0.60206; 


; 2.69897 


; 3.77815; 


; 1.36173 


; 1.65321; 


; 2.51388; 


2.77633 ; 


; 4.77541 


; 2.82816; 



0.84510; 0.95424. 
3.90309; 2.95424. 
1.81954; 1.92428. 
1.86435; 1.96904. 
0.91172; 4.72652. 



611. TABLE OF LOGARITHMS: with Tabular Differekceb. 



N 


O 


1 


2 


3 


4 


5 


6 


7 


8 





432 


42S 


424 


419 


416 


412 


408 


404 


400 


396 


10 


00000 


00432 


(xmo 


01284 


01703 


02119 


02531 


02938 


03342 


03743 




393 


389 


386 


382 


379 


376 


372 


369 


366 


368 


11 


04139 


04532 


04922 


05308 


05690 


06070 


06446 


06819 


07188 


07555 




360 


857 


355 


351 


849 


346 


843 


344) 


338 


885 


12 


07918 


08279 


08636 


08991 


09342 


09691 


10037 


10380 


10721 


11069 




833 


330 


828 


325 


823 


321 


318 


315 


314 


811 


13 


11394 


11727 


12057 


12385 


12710 


13033 


13354 


13672 


13988 


14301 




809 


307 


305 


303 


301 


299 


297 


295 


298 


291 


14 


14613 


14922 


15229 


15534 


15836 


16137 


16435 


16732 


17026 


17319 


289 


287 


285 


2S8 


281 


279 


278 


276 


274 


272 


15 


17609 


17898 


18184 


18469 


18752 


19033 


19312 


19590 


19866 


20140 




271 


269 


267 


266 


264 


262 


261 


259 


258 


256 


16 


20412 


20683 


20952 


21219 


21484 


21748 


22011 


22272 


22631 


22789 




254 


253 


252 


250 


249 


248 


246 


245 


243 


242 


17 


23045 


23300 


23553 


23805 


24055 


24304 


24551 


24797 


25042 


25286 




241 


2:39 


238 


2:37 


235 


234 


233 


232 


230 


229 


18 


25627 


25768 


26007 


26246 


26482 


26717 


26951 


27184 


27416 


27646 




228 


227 


226 


225 


223 


222 


221 


220 


219 


218 


19 


27876 


28103 


28330 


28656 


28780 


29003 


29226 


29447 


29667 


29885 


217 


216 


2 15 


213 


212 


211 


210 


209 


208 


207 


20 


30103 


30320 


30535 


30750 


30963 


31176 


31387 


31697 


31806 


32016 




206 


205 


204 


203 


202 


202 


201 


200 


199 


198 


21 


32222 


32428 


32634 


32838 


33041 


33244 


33446 


33646 


a3846 


34044 




197 


196 


195 


194 


193 


193 


192 


191 


190 


189 


^ 


34242 


34439 


34635 


34830 


35026 


36218 


36411 


35(m 


35793 


36984 




188 


188 


187 


186 


185 


184 


184 


188 


182 


181 


23 


36173 


36361 


36549 


36736 


36922 


37107 


372<)1 


37476 


37668 


37840 




181 


180 


179 


178 


178 


177 


176 


176 


175 


174 


24 


38021 


38202 


38382 


38561 


38739 


38917 


39094 


39270 


39445 


39620 


173 


173 


172 


171 


171 


170 


169 


169 


168 


167 


25 


39794 


39967 


40140 


40312 


40483 


40654 


40824 


40993 


41162 


41330 




167 


166 


166 


165 


165 


164 


163 


162 


162 


161 


26 


41497 


41664 


41830 


41996 


42160 


42325 


42488 


42651 


42813 


42976 




161 


160 


159 


159 


158 


158 


157 


156 


156 


166 


27 


43136 


43297 


43457 


43616 


43775 


43933 


44091 


44248 


44404 


446()0 




165 


154 


154 


153 


152 


152 


151 


151 


151 


150 


28 


44716 


44871 


45025 


45179 


45332 


46484 


46637 


46788 


46939 


46090 




149 


149 


149 


148 


147 


147 


147 


146 


145 


145 


29 


46240 


46389 


46538 


46687 


46835 


46982 


47129 


47276 


47422 


47667 


145 


144 


143 


143 


143 


142 


142 


141 


141 


140 


SO 


47712 


47857 


48001 


48144 


48287 


48430 


48572 


48714 


48865 


48996 




140 


189 


139 


189 


138 


138 


137 


187 


136 


136 


31 


49136 


49276 


49415 


49554 


49693 


49831 


49969 


50106 


50243 


50379 




136 


135 


134 


i;34 


183 


133 


133 


182 


132 


131 


32 


60615 


50651 


50786 


50^)20 


51055 


51188 


61322 


51456 


61587 


51720 




131 


181 


130 


180 


129 


129 


129 


129 


128 


128 


33 


61851 


51983 


52114 


52244 


52375 


62504 


52634 


52763 


52892 


53020 




127 


127 


126 


126 


126 


126 


125 


125 


125 


124 


34 


63148 


53275 


53403 

128 


53529 
128 


53656 


53782 
122 


53S)08 


54033 

121 


64158 


54283 


124 


128 


128 


122 


121 


121 


35 


54407 


54531 


54054 


54777 


54900 


55023 


65145 


55267 


56388 


55509 




121 


120 


120 


119 


119 


119 


119 


118 


118 


117 


36 


65630 


55751 


55871 


55991 


56110 


56229 


56348 


56467 


66586 


56703 




117 


117 


117 


116 


116 


116 


115 


115 


115 


114 


37 


66820 


66937 


67054 


57171 


57287 


57403 


57519 


57634 


57749 


67864 




114 


114 


114 


118 


113 


113 


112 


112 


112 


112 


38 


67978 


58092 


58206 


68320 


68433 


58546 


68659 


68771 


68883 


58995 




112 


111 


110 


no 


110 


110 


109 


109 


109 


109 


39 


59106 


59218 


59329 


59439 


59560 


59660 


69770 


69879 


69988 


60097 



319 



TABLE OF LOGARITHMS. 



N 

40 

41 
42 
43 
44 

46 

46 

47 
48 
49 

50 

51 
52 
53 
54 

55 

56 
57 
58 
59 

60 

61 
62 
63 

04 

65 

66 
67 
68 
(59 


O 


1 


2 


3 


4 


6 


6 


7 


8 


9 


108 
(30206 

106 
61278 

108 
62325 

101 

63347 

99 

64315 


108 
60314 

106 
61384 

108 
62428 

101 

63448 

98 

64444 


108 
60423 

105 
61490 

108 
62531 

101 

63548 

98 

64542 


107 
60531 

105 
61595 

108 
62634 

100 

63649 

98 

64640 


107 
60638 

105 
61700 

102 
62737 

100 

63749 

98 

64738 


107 
60746 

105 
61805 

102 
62839 

100 
63849 

97 
64836 


107 
60853 

105 
61909 

102 

62941 

99 

63949 

97 

64933 


107 
60959 

104 
62014 

102 

63043 

99 

64048 

97 

65031 


106 
61066 

104 
62118 
' 102 
63144 
99 
64147 
97 
66128 


106 
61172 

104 
«2221 

101 

63246 

99 

64246 

96 

65225 


96 
65321 

94 
66276 

92 
67210 

90 
68124 

88 
69020 


96 
65418 

94 
66370 

92 
67302 

90 
68215 

88 
69108 


96 
65514 

94 
66464 

92 
67394 

90 
68305 

88 
69197 


96 
65610 

94 
66558 

92 
67486 

90 
68395 

88 
69285 


95 
65706 

94 
66652 

91 
67578 

89 
68485 

87 
69373 


95 
65801 

94 
66745 

91 
67669 

89 
68574 

87 
69461 


95 
66896 

93 
66839 

91 
67761 

89 
68664 

87 
69548 


95 
65992 

93 
66932 

91 
67852 

89 
68763 

87 
69636 


95 
66087 

98 
67025 

91 
67943 

89 
68842 

87 
69723 


95 
66181 

98 
67117 

90 
68034 

89 
68931 

87 
69810 


87 
69897 

85 
70757 

88 
71600 

82 
72428 

80 
73239 


86 
69984 

85 
70842 

88 
71684 

82 
72509 

80 
73320 


86 
70070 

85 
70927 

88 
71767 

82 
72691 

80 
73400 


86 

70157 

85 
71012 

88 
71850 

81 
72673 

80 
73480 


86 
70243 

84 
71096 

83 
71933 

81 
72754 

80 
73560 


86 
70329 

84 
71181 

83 
72016 

81 
72835 

80 
73640 


86 
70415 

84 
71265 

82 
72099 

81 
72916 

79 
73719 


85 
70501 

84 
71349 

82 
72181 

81 
72997 

79 
73799 


85 
70586 

84 
71433 

82 
72263 

81 
73078 

79 
73878 

78 
74663 

77 
75435 

75 
76193 

74 
76938 

77670 


86 
70672 

84 
71617 

82 
72346 

81 
73159 

79 
73957 


79 
74036 

77 
74819 

76 
76687 

75 
76343 

74 
77086 


79 
74115 

77 
74896 

76 
75664 

75 
76418 

74 
77169 


79 
74194 

77 
74974 

76 
75740 

75 
76492 

78 
77232 


78 
74273 

77 
75051 

76 
75815 

74 
76667 

73 
77305 


78 
74351 

77 
76128 

76 
75891 

74 
76641 

78 
77379 


78 
74429 

77 
75206 

75 
76967 

74 
76716 

78 
77452 


78 
74607 

77 
75282 

75 
76042 

74 
76790 

78 
77525 


78 
74686 

77 
75358 

75 
76118 

74 
76864 

78 
77697 


78 
74741 

76 
75511 

75 
76268 

74 
77012 

72 
77743 


72 
77816 

71 
78633 

70 
79239 

69 
79934 

^s 
80618 


72 
77887 

71 
78604 

70 
79309 

69 
80003 

68 
80686 


72 
77960 

71 
78675 

70 
79379 

69 
80072 

68 
80754 


72 
78032 

71 
78746 

70 
79449 

69 
80140 

67 
80821 


72 
78104 

71 
78817 

70 
79518 

69 
80209 

67 
80889 


72 
78176 

71 
78888 

70 
79688 

68 
80277 

67 
80966 


72 
78247 

70 
T8958 

69 
79657 

68 
80346 

67 
81023 


71 
78319 

70 
79029 

69 
79727 

68 
80414 

67 
81090 


71 
78390 

70 
79099 

69 
79796 

68 
80482 

67 
81168 


71 
78462 

70 
79169 

69 
79865 

68 
80550 

67 
81224 


(57 
81291 

66 
81954 

65 

82607 
64 

83251 
68 

838a5 


67 
81368 

66 
82020 

65 
82672 

04 
83315 

63 
83948 


67 
81425 

66 
82086 

65 
82737 

64 
83378 

63 
84011 


66 
81491 

65 
82151 

64 
82802 

64 
83442 

6:3 
84073 


66 
81658 

65 
82217 

64 
82866 

63 
83506 

63 
84136 


66 
81624 

65 
82282 

64 
82930 

63 
83569 

62 
84198 


66 
81690 

66 
82347 

64 
82995 

63 
83632 

62 
84261 


66 
81757 

65 
82413 

64 
83069 

63 
83696 

62 
84323 


66 
81823 

65 
82478 

64 
83123 

68 
83759 

62 
84386 


66 
81889 

65 
82543 

64 
83187 

63 
83822 

62 
84448 



320 



TABLE OF LOGARITHMS. 



N 

70 

71 
72 
73 
74 



76 

76 
77 
78 
79 

80 

81 
82 
83 
84 

86 

86 
87 
88 
89 

90 

91 
92 
93 
94 

96 

96 
97 
98 
99 


O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


62 
W510 

61 
85126 

60 
85733 

59 
86332 

59 
86923 


62 
84572 

61 
85187 

60 
85794 

59 
86392 

59 
86982 


62 
84634 

61 
85248 

60 
85854 

59 
86451 

58 
87040 


62 
84696 

61 
85309 

m 
85914 

59 
86510 

58 
87099 


62 
84757 

61 
85370 

60 
85974 

59 
86570 

58 
87157 


62 
84819 

61 
85431 

60 
86034 

59 
86629 

58 
87216 


61 
84880 

61 
85491 

60 
86094 

59 
86688 

58 
87274 


61 
84942 

61 
85552 

60 
86153 

59 
86747 

58 
87^32 


61 
86003 

60 
85612 

60 
86213 

59 
86806 

58 
87390 


61 
86065 

60 

86673 

•60 

86273 

59 
86864 

58 
87448 


58 

87506 

57 
88081 

56 
88649 

56 
89209 

55 
89763 


58 

87564 

57 
88138 

56 
88705 

56 
89265 

55 
89818 


58 

87622 

57 
88195 

56 
88762 

56 
89321 

55 
89873 


58 
87679 

57 
88252 

56 
88818 

55 
89376 

55 
89927 


58 
87737 

57 
88309 

56 

88874 
55 

89432 
55 

89982 


57 
87795 

57 
8836(> 

56 
88930 

55 
89487 

90037 

54 
90580 

53 
91116 

58 
91645 

52 
92169 

51 
92686 


57 
87852 

57 
88423 

56 
88986 

55 
89542 

55 
90091 


57 
87910 

57 
88480 

56 
89042 

55 
89597 

54 
90146 

64 
90687 

58 
91222 

52 
91751 

52 
92273 

51 
92788 

51 
932<)8 

50 
93802 

49 
94300 

49 
94792 

48 
95279 


57 
87967 

57 
88536 

56 
89098 

55 
89653 

54 
90200 


57 
88024 

56 
88693 

56 
89154 

55 
89708 

54 
90255 


90309 

54 
90849 

53 
91381 

52 
91908 

52 
92428 


54 
90363 

54 
90902 

53 
91434 

52 
91960 

52 
92480 


54 
90417 

53 
9095() 

91487 
52 

92012 
52 

92531 


54 
90472 

5;^ 
91009 

5;^ 
91540 

52 
92065 

51 
92583 


54 
90526 

91062 

53 
91593 

52 
92117 

51 
J)2634 

51 
93146 

60 
93()51 

50 
94151 

49 
94645 

49 
95134 


54 
90634 

53 
91169 

&S 
91698 

52 
92221 

51 
92737 


54 
90741 

91276 
52 

91803 
52 

92324 
51 

92840 


54 
90795 

58 
91328 

52 
91855 

52 
92376 

51 
92891 


51 

92942 

50 
93450 

50 
93952 

49 
94448 

49 
94939 


51 
92993 

50 
93500 

60 
94002 

49 
94498 

49 
94988 


51 

93044 

50 
93551 

50 
94052 

49 
94547 

49 
95036 


51 
93095 

50 
93601 

50 
94101 

49 
94596 

49 
95085 


51 
93197 

50 
93702 

50 
94201 

49 
94694 

48 
95182 


51 
93247 

50 
93752 

50 
94250 

49 
94743 

48 
95231 


51 
93349 

50 
93852 

49 
94349 

49 
94841 

48 
95328 


51 
93399 

50 
93902 

49 
94399 

49 
94890 

48 
95376 


48 

95424 

48 
95904 

47 
96379 

47 
96848 

46 
97313 

46 
97772 

45 
98227 

45 
98677 

44 
99123 

44 
99564 


48 

95472 

48 
95952 

47 
96426 

47 
96895 

46 
97359 


48 

95521 

48 
95999 

47 
96473 

47 
96942 

46 
97405 


48 

95569 

48 
96047 

47 
96520 

47 
96988 

46 
97451 


48 

95617 

47 
96095 

47 
96567 

46 
97035 

46 
97497 


48 

95665 

47 
96142 

47 
96614 

46 
97081 

46 
97543 

45 
98000 

45 
98453 

45 
98900 

44 
99344 

44 
99782 


48 
95713 

47 
96190 

47 
96661 

46 
97128 

46 
97589 


48 
95761 

47 
96237 

47 
96708 

46 
97174 

46 
97635 


48 

95809 

47 
96284 

47 
96755 

46 
97220 

46 
97681 


48 
95866 

47 
96332 

47 
96802 

46 
97267 

46 
97727 


46 
97818 

45 
98272 

45 
98722 

44 
99167 

44 
99607 


46 
97864 

45 
98318 

45 
98767 

44 
99211 

44 
99651 


46 
97909 

45 
98363 

45 
98811 

44 
99255 

44 
99695 


46 
97955 

45 
98408 

45 
98856 

44 
99300 

44 
99739 


45 
98046 

45 
98498 

44 
98945 

44 
99388 

44 
99826 


45 
98091 

45 
98543 

44 
98989 

44 
99432 

44 
99870 


45 
98137 

45 
98588 

44 
99034 

44 
99476 

44 
99913 


45 
98182 

45 
98632 

44 
99078 

44 
99520 

43 
99957 



321 



WHITB'S ALO. 21 



822 ALGEBRA. [§612. 

APPLICATIONS TO NUMERICAL PROCESSES. 

1. Find the continued product of 875.43, 3.1416, and .00643. 

log 875.43 = 2.04222 
log 3.1416 = 0.49716 
log .00543 = 3.73480 

Sum = 1.17417 = log 14.933, Ans. (§ 690) 

2. Find the quotient of 72.9 by 645.37. 

log 72.9 = 1.86273 
log 645.37 = 2.73670 

Difference = 1.12603 = log .13367. (§ 691) 

3. Find the 60th power of 1.06. 

log 1.06 = 0.02119 

60 



Product = 1.06950 = log 11.468 .... (§ 692) 

.'. 11.468 is the 60th power of 1.06. 

4. Find the 12th root of 366. 

log 366 =2.66229 
Dividing by 12 | 2.66229 

Quotient = 0.21362 = log 1.6350 •••. (§ 693) 

5. Find the 6th root of .0366. 

log .0366 = 2.66229. 
Add to log - 3 + 3, 5 + 3.56229. 
Dividing by 5 (§ 593), 1.71246. 
The antilog of 1.71246 = .61674, the required root. 

612. When, as in Example 6, we have to divide a negative 
characteristic which is not an exact multiple of the divisor, we 
add to the negative characteristic as many negative units as 
will make it such a multiple, and prefix the same number of 
positive units to the mantissa. Thus, above, 2.66229 = 
6 + 3.66229, and, dividing by 6, we obtain 1.71246, the antilog 
of which is .61674, the required root. 



§ 613.] LOGARITHMS. 823 

When negative numbers occur in computations, we proceed 
as if all were positive, deciding the sign of the result according 
to the rule of signs. Thus, to obtain the product of 53.6 and 
— 3.975 by logarithms, we proceed as if both were positive, 
and give the negative sign to the result. 

Find by logarithms the values of the following expressions : 
6. 535 X. 342. ^^ .5673x25.05 



7. 2.937x1.505. 763x365,47 

8. 6.7354 X .8925. ^^' (^•^)'- 

9. 12345 X .0053782. ^^' (1.0546)'®. 

10. .075 X .3678 ^ 73.251. ^®- (-^054 x .53798)* 

56.32 8275.5 ^'^' V§J^ 

' 9.365* * 43296* 18. a/1.0562. 

613. Exponential equations (§ 569) are most readily solved 
by the aid of logarithms. When, for example, the a, I, r, of a 
geometrical progression, are given to find n, the value of n 
must be found from the equation lz=ar^~^] whence r*~' = Z -5- a. 

Taking the logarithms of both members of this equation, 

(n — 1) log r = log I — log a • 

whence n = 1 + ^^S I -log a . 

logr 

By taking from the table the logarithms of the given quan- 
tities a, I, r, and performing the indicated operations, we obtain 
the value of n. If in the foregoing formula we suppose 
a = 3, 1 = .00019683, r = .3, we obtain 

n:=l \ ^Qg -00019683 - log 3 

'^l i^^:3 ' 

^ , 4.29409 - 0.47712 
or =1H = ; 

1.47712 

whence » = 1 + |§1^= 1 + 8 = 9. 

1.47713 



824 ALGEBRA. [§ 614. 

19. Given a = 162, i= ^, r = J: find n. 

20. Given a = ^, « = 4o, r = 4: find n. 

21. Given a = 343, « = 400|, / = | : find n. 

22. Given r = 5, «=1562, Z = 1250: find ». 

Solve the exponential equations 

23. 11« = 1331. 26. 13' = 14. 29. 4'-« = 6. 

24. 5** = 15625. 27. 17' = 91. 30. 3'+* = 2187. 

25. 12' = 41.57. 28. alf = c. 31. 3''^+* = 1200. 

(iS^ee also § 689.) 

BUSINESS FORMULAS. 

614. Problems in percentage can in most cases be readily 
solved by the methods of arithmetic ; but even in arithmetic 
there may be advantage in the use of general formulas to in- 
dicate the processes involved.* Such formulas have the 
advantage of being so related that they can be derived from 
each other, and thus be readily reproduced when needed. In 
algebra the solutions of all percentage problems may be indi- 
cated by formulas with the further advantage that they permit 
the easy use of logarithms, thus greatly facilitating computa- 
tion. 

Formulas for the more common processes in percentage, in- 
cluding simple interest, have already been given (§§ 285-289); 
and all that is further needed is the presentation of general 
formulas for compound interest and annuities, with examples 
illustrating the use of logarithms. 

Compound Interest. 

616. The formula for finding the amount of any given 
principal at compound interest for any time and any rate per 
cent may be obtained as follows : 

Let p denote the principal, r the rate per cent, and n the time 

* For methods of using percentage formulas in arithmetic, see Whitens 
New Complete Arithmetic. 



§ 616.] LOGARITHMS. 825 

in years. Since the amount of one dollar in one year is 1 + r, 
the amount of p dollars in one year is pil-^-r) ; and the amount 
of p (1 -h r) dollars in one year is 

i)(H- r) X (1 + r)=:i)(l + ^)', 
which is accordingly the amount of p dollars in two years. 
The amount of p(l -{■rf dollars in one year is 

p(l + r)«x(l + r)=i>(l + ^)', 
which is the amount of p dollars in three years; and so on. 
Hence the amount of p dollars in n years is p(l + r)" ; that is, 
denoting the amount by a, 

a=p(l-fr)». (B^ 

616. Any three of the four numbers denoted by a, p, r, w, 
being given, the fourth may be found from the above formula, 
which therefore suffices for the solution of all cases in com- 
pound interest. 

If a, p, r, are given to find n, we have to solve the exponential 
equation 

whence n log (1 + r) = log a — log jp, 

and n = (log a — logp) -^ log (1 + r). 

1. In how many years will $ 365 amount to $ 500 at 4 % 
compound interest ? 

Here a = 600, p = 365, and 1 + r = 1.64. Substituting the logarithm 
of these numbers for those of a, p, 1 + r, above, we find 

n =(2.69897 - 2.66229) -r- .01703 = 8.026. 

2. At what rate per cent will $ 1500 at compound interest 
amount to $ 1889.568 in 3 years ? 

3. What principal will amount to $ 535.9572 in 3 years at 
6 % compound interest ? 

4. Find a formula to show how long it will take p dollars to 
amount to mp dollars at r % compound interest. 



326 ALGEBRA. [§ 617. 

617. If the interest is payable at shorter intervals than 
a year, say every half year, then, at r % per annnm, the 

interest on one dollar for half a year is - ; and at compound 
interest the amount of p dollars in n years, that is, in 2 n half 
years, is jp[l+^] , the same as the amount of p dollars in 2 n 

years at ^%. In the same way, if interest is payable m 

times a year, the amount of p dollars at r % is expressed by 
the formula 



-(• ^0 



Ankuities. 

618. To find the amount of an annuity accumulating any 
number of years, allowing compound interest,* 

Let a denote the number of dollars in the annuity, n the 
number of years, 1 4- r the amount of a dollar for one year, 
and A the required amount. At the end of the first year a is 
due, and amounts at the end of the second year to a (1 + r) ; 
hence the whole sum due at the end of the second year is 
a -h a(l -I- r) = a[l -f (1 -f- r)]. At the end of the third year the 
sum due is, in like manner, 

a[l + (l-hr) + (l4-r)(l + r)] = a[l + (l + r) + (l + r)«]. 

By proceeding in this way, we find that the amount due at the 
end of n years is 

a[l + (l + r) + (l + r)^ + (l + r)»+...+ (l + r)-^], 

a G. P. whose first term is a, and ratio 1 + r ; hence we 

obtain A = a ^] +^)'* "} = - [(1 + rY - 11. 

* Annuities at simple interest have practically no existence. 



§ 620.] LOGARITHMS. 827 

619. To find the present value of an annuity to continue a 
certain number of years, allowing compound interest, 

Let p denote the present value. The amount of p in n years, 
that is, p(l -f r)", should be equal to the accumulated amount of 
the annuity in the same time ; hence 

^=-:('-(iir) <" 

If we suppose n to be indefinitely great, that is, the annu- 
ity to be perpetual, then — — becomes — =0, and the forego- 
ing formula becomes p = — (2) 

620. Besides problems relating to annuities proper, these 
formulas may be applied to the solution of a great variety of 
problems concerning life insurance, value of estates, increase of 
capital, population, etc. 

Problbms. 

1. How much should a man pay down to obtain a life 
annuity of $ 1000, his expectation of life being estimated at 20 
years, and interest reckoned at 5%? 

Here a = 1000, n = 20, r = .05. Substituting these values for o, n, r, 
in Formula (1), we obtain 

_1000/j 1 \ 

^ .05 V (1.05)»A 

By the aid of logarithms we find ^ = .3769... ; 

hence p = ^522(i _ .3769) = 12462. Ans. $ 12462. 

.05 

2. What is the value of a farm that yields a net rental of 
$ 900, interest reckoned at 6%? 



328 ALGEBRA. [§ 620. 

Here we hftve to find the present yalue of a perpetoal annuity of $ 900. 
By Formula (2) above, we obtain 

900 __ jg^jQ^ ^^ $15000. 

^ .06 • 

8. What is the present value of an annuity of $ 600 for 25 
years, interest reckoned at 5%? 

4. What sum should be paid for a 15 years' lease of a prop- 
erty yielding a net annual profit of $ 500, interest at 6% ? 

6. By giving up smoking, a man saves $ 50 a year. How 
much does he thus save in 30 years, interest at 5% ? 

6. A man with a capital of $ 10,000 spends every year $ 150 
more than his income. In how many years will his capital be 
consumed, interest reckoned at 8%? 

7. A population of 1,000,000 has a steady annual increase of 
3%. In how many years will it double itself ? 

8. A man with a capital of $ 20,000 adds to it yearly $ 200 
besides the interest. What will be the amount of his capital 
in 20 years, interest reckoned at 10%? 

9. Having borrowed $12,000 at 5%, how much should 
I pay every year so as to discharge the debt in 10 years ? 

10. In order to accumulate $ 3000, to cover the expenses of 
a son's college course, a father invested a certain sum at 5%, 
on each recurring anniversary of his son's birth until the 18th 
inclusive. How much did he invest yearly ? 

11. What annual payment will amount to $5000 in 21 
years at 6% compound interest ? 

12. What is the value of an estate yielding $1500 net 
annual income, interest reckoned at 5^%? 



§622.] UNDETERMINED COEFFICIENTS. 329 



CHAPTER XXI.* 
UNDETERMINED COEFFICIENTS AND APPLICATIONS. 

621. It is sometimes desirable to obtain for a given alge- 
braic expression an equivalent of a certain form ; as, for ex- 
ample, a series in ascending powers of x. To this end, we 
first assume that the given expression is equal. to a series of 
the required form, but having undetermined coefficients. We 
then proceed to determine the values of these coefficients by 
means of the principles exemplified below. 

622. (1) If an equation of the form 

is such that for every value of x the equation is an identity 
(§ 624), then the coefficients of the like powers of x in the two 
members are equal. 

For, since the equation is satisfied by any value of «, let 
a; = 0. Then every term containing x equals 0, and the equa- 
tion reduces to 

Subtracting these equals from the members of Equation (1), 
we have 

Bx+C7?^D7?'{- ... = B'x 4- (7'aj2 -|- D'oi? + ... (2) 

Dividing each member of (2) by x, we have 

JB + Ca: + Z>a^ -f - = ^' + O'a; + D'x" + .... (3) 

* This and the following chapters are designed for more advanced 
classes, and especially for stiMents who are preparing for higher institu- 
tions who^e entrance requirements may include an elementary knowledge 
of one or more of the subjects treated. 



330 ALGEBRA. [§ 623. 

Since Equation (3) must be satisfied by any value of a?, let 
a; = ; and then the equation reduces to 

In the same way we can prove that C = C, D=D\ and so on. 

623. (2) If an equation of the form 

^-h5a?-hOc' + l>«'+ — =0 (4) 

is true for every value of a?, the coefficients must each eqtuil zero. 

For, since x may have any value, let a? = 0, and then the 

equation reduces to 

^ = 0. 

Then omitting A in (4), and dividing by a?, we have 

J5 + (7a; + 2>ar*+... =0. (5) 

Since x may have any value in (5), let a? = 0, and then we 
obtain B = ; and in the same way we can prove that C7= 0, 
D = 0, and so on. 

624. It can be shown that the foregoing principles (§§ 622, 
623) apply only to equations that contain convergent series 
(§ 631, note), but the proof belongs to higher algebra. 

Let us now consider several of the more important applica- 
tions of these principles. 

RESOLUTION OF FRACTIONS. 

625. To resolve a fraction into partial fractions is to find the 
fractions whose algebraic sum is the given fraction. This is 
the converse of the process of adding fractions given in § 264 

For example, -^ ^ = -^^ r^; and conversely, 

^ x — b a;-f3 or — 2aj — 15 

the fraction „ ^'^^^ ^^ being given, it is required to find 

a^ — 2 a; —• 15 

the partial fractions of which this given fraction is the alge- 
braic sum. 



§626.] UNDETERMINED COEFFICIENTS. 331 

Having factored the denominator, we assume that 



x'-^2x-15 x + 3 x-5' 

A and B being the numbers which we wish to determine. 
Clearing (1) of fractions, we obtain 

x + 19 = Ax-5A + Bx + 3B, 

or a; 4- 19 =(^ -f B)x-(5A - SB). 

Equating the coefficients of the like powers of x (§ 622), we 

obtain 

A + B=l, (2) 

5J.-35 = -19; (3) 

two equations each containing A and B, from which we obtain 

^ = -2, 5 = 3; 

whence — = • 

ar^-2aj-15 x-5 flj + 3 

626. Suppose, however, that it be required to resolve into 

partial fractions the expression ^ ~ ^^« 

or — 2 X — 15 

If, having factored the denominator, we should assume that 

2«2-13a;-9 A B 



x'-2x-15 a; + 3 aj-5' 

we would be led, on clearing of fractions, to the absurd result 
2 = 0. This difficulty may be avoided by first reducing the 

fraction — — ^^^- — ^"^ to a mixed number, and then resolving 
or — 2 X — 15 

the fractional part, as shown below. 

Hence, whenever the numerator of the fraction to be resolved 

is not of lower degree than the denominator, reduce the given 

fraction to a mixed number, and then resolve the fractional part. 



332 ALGEBRA- [§ 627. 

Thus, taking the fraction given above, 



. -9xH-21 A . B 

Assume -— -^ — • = -\- 



(a; + 3)(a;-5)~aj-|-3 x-5' 
then, clearing of fractions, 

■- 9 X -\-21 = Ax- 5 A-\-Bx-\- SB = (A-^B)x -(5 A — 3 B) ; 
whence, equating, A-^ B = — 9] 5^ — 35 = — 21; 
whence A = — 6, 5 = — 3; 

whence H^zl^^^^2- ^ ^ 



a^-2x-15 x-\-3 x-5 

Resolve into partial fractions 

2a;-13 , 2a^ + g4-3 

aj2 _ 13 aj -f 40' ' a^-1 

2a;4-15 « 3a^4-3a^-f-2 

ar^ — 15 ic + 56 a? — x 

7a?-23 ^ 2 a^ -f 21 a; + 13 

ar«_6a; + 5" ' 4a^-5a^-hl ' 

4a? -29 jQ 3a^ + 3a;H-i8 

ar^ + 3a;-10* ' a^-9x 

13 a;+g ji 8a; -1-12 

6a^^5x-\-l' ' a^-\-6x-\-S 

2a^ + 2x-6 ,„ 2«2 + a:-l 

aj34.5aj24.6aj 2a;2^^_3 

EXPANSION OF FRACTIONS INTO SERIES. 

627. (1) Let ^-^^^ ^ be the given fraction. 

1 — a? — ar 

Assume ."^"^^^ , = A-{- Bx-\- Ca^ -{- Di^ -{- '"- (1) 

1 — a; — ar 



1. 



2. 



3. 



4. 



5. 



6. 



§628.] UNDETERMINED COEFFICIENTS. 333 

Multiplying each member of (l)hj 1 — x — oi^, we obtain 

l + 2x = A-^B x-\-C a^-i-D aj»+etc. 
-A 



x-\-C a^-i-D 


-B 


-0 


-A 


-B 



Whence A = l', B-A = 2, .-. ^ = 3; (7-(^ + J5)=0, 
.-. 0=4; D-(5+O)=0, .-. i> = 7; etc. 

Substituting these values in the assumed series (1), we 

obtain ^i±l^ = l + 3a; + 4ic2 + 7ar^ + ll«*4-18a^+..., in 
1 — a? — ar 

which each coefficient after the second is equal to the sum 

of the preceding two coefficients, and hence the series can be 

continued indefinitely. 



628. (2) Let — ^^ — ^^ be the given fraction. 
^^ aj2-3a;-|-2 ^ 

We see by inspection that the first term of the development 
must contain x~^\ and we accordingly assume that 

^-^^ =^-2 + J5a;-i + + i>a+ — , (1) 



aj*-3a?-h2 

and proceed as above. 

Similarly, in other cases, we should first determine by in- 
spection what power of x must occur in the first term of the 
development, and then assume the series accordingly. 

1 — 2r 1— 2rr 

The fraction — - — =-=^ — may be written in the form — , and 

then developed as in (1) in § 627. 

Such developments as those here referred to may be obtained by three 
distinct methods: to wit, (1) by simple division^ which may be em- 
ployed as a means of proving the correctness of the results obtained 
by other methods; (2) by the method of undetermined coefficients^ as 
exemplified above; and (3) by means of the binomial formula ^ as 
shown §§ 630, 631. 



384 ALGEBRA* [§ 629. 

Expand into an infinite series 

1. UM^. 6. ' 



2. -A 1- 6. 



l+3a; 






1 




1 


-2aj 


+*• 




1- 


X 


1 


-2a; 


-3a? 




1 +a; 





3. ^ 1. — n__.. 7. 



4- . "^"  8. 

1-aj + a? 



BINOMIAL FORMULA. 



3-a; 


l-\-x 


l-2x-a? 


14- a; 


l+2a;4-3a;* 


a-hbx 


l+2a; + 3a? 



629. It has been shown by successive multiplications of the 
binomial a + & (§ 321), that, if n denotes any positive integer 
from 2 to 5 inclusive, 



(a + by = a* + na^-'b + '^^'^ "J^^ a^-^'b' 

1 • ^ 

, w(n — l)(n — 2) ^_3,3 , ,^. 

+ ^ 1.2.3 ^« ^M— •. (1) 

It remains to prove that this formula holds true for any 
positive integral value of n (§ 324). 

Assume that Formula (1) is true for any positive integral 
value of w. Multiplying both members of (1) by a 4- 6, we have 

(l)x(4-&) a»64- na»-^624- !L^-:^Da"-V4— • 

1 • i6 

Collecting terms, we obtain 

(a 4- 6)""^' = a"-'' 4- (w 4- 1) a^h + i^±I)^a-W 

1 • 2 



§630.] UNDETERMINED COEFFICIENTS. 886 

Comparing (1) and (2), we see that the same law holds in 
both for the formation of the coefficients, and also for the 
exponents ; and hence, if the formula holds true for the expan- 
sion of (a -f- by, it holds true for the expansion of (a 4- by^\ 
But it has been shown by actual multiplication that Formula 
(1) holds true when n = 5 (§ 321), and hence it holds true 
when n = 6 ; and, if it holds true when n = 6, it holds true 
when n = 7, and so on indefinitely to any power denoted by 
a positive integral exponent 

Again, if 6 is a negative number, then, since the odd powers 
of a negative number are negative, and the even powers positive 
(§ 318), we evidently have, from (1), 



(a + by = a» - na^-^b + ^^^"^^ a^-'y 



w(n — l)(n — 2) ^ .,3 , ,^. 
1.2.3 ^a* '&* + —, (3) 

the alternate signs being -f- and — . 
Equations (1) and (3) can be written as one formula as follows : 



(a ± by = a- ± wa«-*6 + ^^^ "^''"^ «""^^' 

1 • 2 



^ 1.2.3 — ^* ■*■ ***• ^^ 



Negative or Fractional Exponents. 

680. It can be shown that the above formula (A) holds also 
for both negative and fractional exponents; but the results 
obtained are reliable only when the resulting infinite series is 
convergent. 

When the exponent n is a positive integer, the series ter- 
minates with n + 1 terms ; for the coefficient of the next and 
each succeeding term contains the zero factor n — n, which 
causes the terms to vanish. But when n is negative or frac- 
tional, no factor can become zero, and hence the series will 



386 ALGEBRA. [§ 631. 

never terminate, but is an infinite series, either convergent or 
divergent. 

631. The eonvergency and divergency of series is too diffi- 
cult a subject for consideration in an elementary algebra, and 
the same is true of a perfectly rigorous demonstration of the 
binomial formula for negative and fractional exponents.* 



* None of the proofs given in the ordinary algebras are free from 
objections ; and only a few of the more recent make any reference to 
failure of the formula in the case of divergent infinite series. 

We here add a few notes respecting the eonvergency and divergency 
of series to indicate the difficulty of the subject. 

A series consisting of an infinite number of terms, which succeed each 
other according to some fixed law, is said to be convergent when the sum 
of its first n terms approaches nearer and nearer to a finite limiting 
value, according as n is taken greater and greater; and this limiting 
value is called the sum of the series, and from this value the series can 
be made to differ by an amount less than any assigned quantity on taking 
a sufficient number of terms. 

If the sum of the first n terms approximates to no finite limit, the 
series is said to be divergent. 

Let us illustrate by a few examples. 

Take -J— = l-hx + x^-\-x^ + .... 

1 — X 

If X < 1, this series approaches a fixed limit. If, for example, x = J, 
then we have 

' =3^1 + 1+1+1+.... 



1 - J 2 3 32 38 

The greater the number of terms taken, the nearer the sum approaches 
the fixed limit |, and similarly for any value of x less than 1. Hence 
the series is convergent if x < 1. 

On the other hand, if x > 1, the series is divergent. For example, let 
a; = 2, and then we have the manifestly absurd result 

-J— = -1=1 + 2 + 4 + 8 + .... 
1-2 

In this case the sum does not approach a fixed limit, and the series is 
therefore divergent: 

Again, let —1— = l - x + x^ - sfi + .... 

1 +» 



§631.] UNDETERMINED COEFFICIENTS- 337 

We shall therefore assume the formula to be true for such 
exponents in cases in which the values of a and b are such as 
to result in convergent series; and, in the examples given 
below, care will be taken to indicate when the resulting series 
is convergent. 

1. Expand (a + h)-'\ 

Here n=-l ; ^^ =^:i^ = -l ; ?^ = '- 1; ?^ = -i±^ = -l. 

It is thus seen that the coefficients are alternately + 1 and — 1, since 
the product of an even number of negative factors is +« a^d of an odd 
number of factors — (§ 106). 

Hence 

(a + 6)-i = a-i - a-^b + a^^b^ ... + a-'6'-i = -(1-- + ^--+-), 

o \ a Qi Cb I 

a series which is convergent when a > &. 

It is often found convenient, as in this example, to place a factor out- 
side a parenthesis, and to change negative to positive exponents. 

2. Expand (a + &)* 

Using formula A^ b being positive, we have 

(a-f 5)i=ai+ 1 a-^b- ^ ah^^ ^ ah^- ^^^ a'hi^... 

V 2a 2. 4a2 2.4. 6a8 2.4.6.8a* / 

This series is also convergent when a>b. 

It is usually well, as in this series, to keep the factors of the coefficients 
separate, so as to show what is termed the law of the series. 

Here, if x < 1, the sum of the series approaches a fixed limif, and 
hence is convergent. 

If a; = 1, then the series becomes 1 — 1 + 1 — 1 + •••. For an even 
number of terms the sum is ; but for an odd number of terms the sum 
is + 1. In this case the sum oscillates from to + 1 without approach- 
ing a fixed limit, and hence the series is divergent. 

If X > 1, the sum of an even number of terms of the series is always 
negative, and the sum of an odd number of terms is always positive. 
Hence, in this case, the sum does not approach a fixed limit, and the 
series is therefore divergent. 

WBITS'S ALO. — 22 



888 ALGEBRA. [§ 632 

8. Expand or (a -f &) . 

Here«--J,-y----^- 4' 3 " 6'^" g,aiia800ii 
Hence (a + 6)"* = a"* - 5 «"' & + ^^ «"**^ - ^"4^ «"^^» + - 

V^V 2o 2.4a« 2.4.6a« J* 
This series is convergent when a > 6. 

Expand to four terms 

4. (1 + iV)'* 7. (2 + i)* 10. (a + 6)-« 

6. (1-i)"*. 8. (1+a)* 11. (a-6)~*. 

6. (1+i)* 9. (l-a)l 12. (1 + i)*. 

The series in 8 and 9 is convergent when a < 1, and in 10 and 11 
when a > &. 

Extraction op Roots by the Binomial Formula. 

632. The binomial formula may be employed for the extrac- 
tion of any roots of a given number N. 

For ^-^= Va**±6 = a[l ± — )«, a* being, the nth powei 
which is near to N, either greater or less than N\ but — must 



a" 



be a proper fraction, otherwise the expansion will be a 
divergent series, and the result will not give the correct root. 

When (1 )» is expanded, the number of terms required 



a" 



to give a certain degree of approximation will depend upon the 
relative value of a** and 6. 
As an example, let us find the square root of 72. 

V72 = V64+8 = 8 (1 + 1)*, since |; = ^ = |. 



§633.] UNDETERMINED COEFFICIENTS. 889 

Expanding (1 + \y by Formula (A), we have (since a = 1, 
b = l, and n = ^), 



\ Sy V 2 8 2.4-8' 2.4.6. 



8» 
1.3.6 



+ 



•) 



2.4.6.8.8* 

Performing the operations indicated^ we obtain 

V72 = 8.48528, 

which is correct to the fifth decimal figure. 

But few terms are necessary in this example, since a" is 8 times 6, and 
the convergence of the series is rapid. 

1. Extract the cube root of 17. 

Here the third power nearest to 17 is 8, whence 

v^ = -y/sT^^ 2 (1 + f)i. 
But since -^ is an improper fraction, the expansion of (1 + |)^ wi]\ 
give a divergent series^ and the result will not be a correct value of ^17. 
But since y/vi = y/27 - 10 = 3 ^1 - J?, we can write 

^=3fi-15\* = 3(i-^l?-^fl5V--lil-fl5V 

V 27/ V 3 27 3.6\27y 3.6.9^27^ 

2.6.8 /loy \ 
3.6.9.12\27/ ** /' 

This is a convergent series, and, by taking a sufficient number of terms, 
we can find the value of vTf to any required approximation. The con- 
vergence of the series is, however, not rapid. 

633. The following examples will converge rapidly, and but 
a few terms will be required to obtain results correct to five or 
six places of decimals. 

2. ■\/2S. 3. a/34. 4. \/i29. 6. </244. 

This process of extracting the roots of numbers, though interesting as 
an application of the binomial formula, is practically of little value when 
logarithmic tables are available. 



840 ALGEBRA. [§631 



CHAPTER XXII. 
DETERMINANTS. 

634. An expression whose value depends on two or more 
quantities is said to be a function of those quantities. Thus, 
a6 -h 6c is a function of a, 6, and c (§ 640). 

In solving a system of equations of the first degree, also 
defined as linear, a class of functions appear, called deter- 
minants, first observed by Leibnitz in 1693. 

635. Let us take the equations 

faia; + 6,2^ = 0, (1) 

(a2a; + % = 0. (2) 

Multiplying (1) by h^ and (2) by — 6^ adding, and then 
dividing the resulting equation by x, we have 

ai62 — o,j^i =5 0. 
The expression afi^ — ajb^ is a determinant It is usually 
written with vertical lines at the left and right, called the 

«gware/orw, thus: ^ ,\ Hence ^ ^ =0^2 — 0^1* 



636. The first member of this identity is the undeveloped 
form of the determinant, and the second member its developed 
form. 

The numbers ai, 02, 61, and b^ are called the elements of the 
determinant, and the expressions 0162 and 0261 in the developed 
form are called its terms. 

The horizontal lines of letters in the undeveloped form of a 
determinant are called rows, and the vertical lines of letters 
are called columns. 



S 639.] 



DETERMINANTS. 



341 



The determinant 






has two rows and two columns, and, 



in the developed form (afiz — aj)i) each term is the product of 
two factors ; hence the determinant is said to be of the second 
order. 

It will be seen later that in every determinant there are as many rows 
as colunms, and as many of each as there are elements in each term. 

637. It will be observed that the developed form of the 



determinant 



^2 62 



is the product of the elements in the 



diagonal from the upper left-hand corner to the lower right- 
hand comer (afi^ minus the product of the elements in the 
diagonal from the lower left-hand corner to the upper right- 
hand comer (a^ bi). 

The first of these diagonals (ai b^ is called the principcd 
diagonal, and the second (0261) the secondary diagonal. 



638. Since 



Oi bi 
02 ^2 



= afii — djbi = afii — bia^ it follows that 



bi 62 



Hence the value of the determinant is not altered by 
changing the rows into columns, and the columns into rows, 

639. Again, since afi^ — ajbi = — (biO^ — ftjOi) = — (ajbi — arjb^, 
it follows that 



ttl 61 




61 Oi 




02 62 


a, 62 




&2 <h 




Oi bi 



Hence, if the order of the columns or rows be changed, 
the sign of the determinant wiU be changed, but its absolute valve 
will not be altered. 

It will be seen later (§ 654) that the above laws hold true for a deter- 
minant of any order. 



842 



ALGEBRA. 



r§640. 



640. Let us now solve the simultaneous linear equations 

(ai^ + bjy = Ci, (1) 

( ojaj + 6^ = Cj. (2) 

Multiplying (1) by &s and (2) by —hu and adding the resulting equa^ 
tions, we obtain 

Multiplying (1) by — a,, and (2) by Oj, and adding the resulting equa- 
tions, we obtain 

It will be observed that the numerators and the denominators in the 
values of x and y are determinants, the denominators being alike, and 
hence we can write 



x = 



Cl 


6. 


c. 


h 


«1 


6. 


0, 


6. 



(3) 



V = 



<h 


<H 


a, 


c« 


«i 


&i 


a, 


ft. 



(4) 



Comparing the numerators and denominators, it will be seen 
that the numerator of (3) can be obtained from the denomina- 
tor by changing the column of a's in the denominator to &s ; 
and, in like manner, that the numerator of (4) can be obtained 
from the denominator by changing the column of I/a in the 
denominator to c's. 



EXBRCISBS. 

1. Solve, by means of (3) and (4), the equations 

3aj + 4y=17. 

Here ai = 5, as = 3 ; &i = 2, &2 = ^ ; ci = 19, and Ci = 17. Substitut- 
ing these values in (3) and (4), we obtain 



{ 



« = 



19 2 
17 4 


76-34 o 
= 20-6=^' y = 


6 19 
3 17 


86-67 


6 2 
3 4 


6 2 
3 4 


~ 20-6 



= 2. 



§642.] 



DETERMINANTS. 



848 



Solve in like maimer the following linear equations : 
2. 



3 (4aj-y = 18, 
\2x-\-3y = 16. 



\5x-2y = 3. 

ax + by = c, 
dx + cy =/. 



5 



•{ 



Expand the following determinants : 



6. 



5 
4 



3 
2 



7. 



15 
3 




5 



8. 



4 
3 



5 




DETERMINANTS OF THE THIRD ORDER. 

641. Let us take the equations 

aix 4- &iy + Ci« = 0, (1) 

a^-\-b2y-\-c^ — 0, (2) 

a^-\-b^-{-c^ = 0, (3) 

MultiplyiDg (1) by 62C8 - 68C2, (2) by - biCz + bzCi, and (3) by 61C2 
— b^u suid adding the resulting equations, thus eliminating y and z, and 
then dividing the resulting equation by x, we obtain 

ai(6aC8 - &8Ca) - aaC&iCa - fesCi) + azipic^ - b^i) = 0, 



or 



ai 



62 Ca 


— 02 


61 Ci 


+ fl8 


61 Ci 


68 Cs 




bs cs 




62 ca 



= 0. 



(4) 
(6) 



If the operations indicated in (4) be performed, we have 



62 C2 


-02 


61 Ci 


+ 08 


61 Ci 


68 Cs 




6« C8 




62 C2 



(6) 



642. The first member of (6) is a determinant ; and, since 
each term contains three demerUSj it is called a determinant of 
the third order. 

Such a determinant is usually written in the square form, and hence 
(6) may be written 



(7) 



a\ 61 c\ 




62 C2 




61 Ci 




61 Ci 


02 O2 C2 


= 01 


6s cs 


-Oa 


6t ct 


+ 08 


b% 0% 


Ot 08 Cg 















344 



ALGEBRA. 



t§«4d. 



«1 


bi 


Cl 


a2 


62 


ca 


«3 


bz 


cs 


ai 


bi 


Cl 


a2 


b2 


d 



643. A determinant of the third order may be readily 
developed as follows: 

Repeat the first and second rows below, as at the right 
For the positive terms, begin with m, «2» «8» respectively, 

and multiply the elements diagonally downward, obtaining 

€iib2Ci, ozbzCi, asbic^. 

For the negative terms, begin with as, ai, 02* respectively, 
and multiply the elements diagonally upward^ obtaining 
ttsbifiij aibzc^i a2&iC8. 

The development is aibfps + OzbsCi + asbii^ — azb2Ci — aibsC2 — aaftiCg. 

In practice it will not be necessary to write down the repeated rows, 
for the work can easily be done mentally. 

644. If we change the rows to columns, and the columns to 
rows, we shall obtain the same result. Thus, 

ai 02 az 

= aib2Cz + biC2as 4- CiOa&s — fliC2&8 — bia2Cs — CiboQz' (8) 



61 62 bs 



Cl C2 Cz 

It thus appears that the value of a determinant of the third 
order is not altered by cTianging the rows to columns, and the 
columns to rows. 

645. If the order of any two rows or columns be inter- 
changed, the result obtained will be the negative of (8). 
Thus, 

02 Oi as 

&2 bi bz = 02&1C8 + 62Cifl8 + C2O168 — Ca^ids ~ b^fliCs — a^^bgci. 

Ca Cl C8 

It is thus seen, that, if any two columns or rows of a deter- 
minant of the third order be interchanged, ths sign of the 
determinant will be changed, but its value will not be altered, 

646. It appears from the first member of (6), § 641, that a 
determinant of the third order consists of six terms, three of 
which are positive, and three negative. 

It is seen from (7) that a determinant of the third order 
can be resolved into the sum of three determinants of the second 
order. 



647.] 



DETERMINANTS. 



345 



EXBRCISBS. 



Develop the following determinants : 



1. 



2. 



3. 



3 


2 


5 




1 


3 


2 


= -24. 


4 


1 


3 




5 


1 


4 




3 


2 


5 


=15. . 


4 


-1 


2 




8 


1 


6 




3 


6 


7 


= -360. 


4 


9 


2 






7. 


SI 


iiow that 



4. 



5. 



6. 



8 4 5 
2 3 1 
5 2 6 



1 -2 

2 6 



-4 



a 
b 
c 

a 
1 
2 



1 
3 
2 

3 
4 

8 



3 

2 
2 



=86. 



2 
4 
1 



=36-5a-2c. 



c 
2 
1 

2 5 



= what ? 



4 3 2 

5 16 



647. Let us now solve the linear equations 



(1) 

(2) 
(3) 



Multiplying (1) by &2C8-&8C2, (2) by -ftiCs+ftsCi, and (3) by 61C2-62C1, 
and adding the resulting equations, the terms that contain y and z vanish, 
and we obtain 



x = 



dih^Cfi — d\bzC2 — d^biC^ + d2Jb%c\ .+ d^bic^ — ^3^2^! 

   .1.   I i_ - - .. -  , I ^ 

ai&2C8 — a\hzC2 — a^biCz + 02^8^1 + azhic^ — azb^\ 



(4) 



Similarly, multiplying (1) by — 0^10% + azC2i (2) by aiCs - azCu and 
(3) by — a\C2 4- «2Ci, and adding the resulting equations, the terms that 
contain x and z vanish, and we have, after ieirranging the terms, 



y = 



01^2^8 — dxdzCj — a^dxCz + ciidzC\ + (izd\c^ — g8d2Ci , 
ai&flCs — ai&8C2 — 02&1C8 + oa&aCi + fls&iCa — as&aci 



(6) 



846 



ALGEBRA. 



[§647. 



LasUy, multiplying (1) by a^&s — Os&a, (2) by — aibz + as&i, and (3) 
by ai&s — fH^u and adding the resolting equations, the terms that con- 
tain X and y vanish, and we have, after rearranging the terms, 



Z r=  ' 



(6) 



The numerator and the denominator in the above values of x, y^ and z 
respectively are determinants of the third order, and they may be written 
in square form, thus : 



X = 



di 


bi 


Cl 


di 


&2 


C2 


dz 


bz 


<58 


«i 


bi 


Cl 


0? 


&2 


C2 


az 


68 


C8 



(40 » = 



ax 


dx 


Cl 


Oi 


d2 


C2 


az 


dz 


cz 


«i. 


bi 


Cl 


02 


62 


Ci 


az 


bz 


Cz 



(5') 



z = 



ai 


&i 


<il 


02 


&2 


d2 


az 


68 


dz 


Ol 


61 


Cl 


02 


&2 


Ct 


as 


&8 


Cz 



(60 



It is seen that the denominators in these values of x, y^ 
and z are the same; and that the numerator of the value of 
X can be obtained from its denominator by changing the column 
of a's in the denominator to d's; that the numerator of the 
value of y can be obtained from its denominator by changing 
the column of &'s in the denominator to d's; and that the 
numerator of the value of z can be obtained from its denomi- 
nator by changing the column of c's in the denominator to d's. 

rSa; -f4y-|-22; = 17, 
8. Solve the linear equations -<5a;-f ^ + 32 = 16, 

Uaj-f 3y-|-72 = 31. 

Here ai = 3, 02 = 5, as = 4 ; 61 = 4, 6a = 1, 69= 3 ; Ci = 2, C2 = 3, 
cg = 7 ; (?i = 17, ^2 = 16, dz = 31. Substituting these values in (4'), (6'), 
and (6'), we obtain 



X = 



17 


4 


2 


16 


1 


3 


31 


3 


7 


3 


4 


2 


5 


1 


3 


4 


3 


7 



_ 119 + 96 + 372 ~ 62 - 448 - 163 
21 + 30 + 48-8-140-27 



-76 
-76 



= 1. 



§648.] 



DETERMINANTS. 



847 



y = 



3 


17 


2 




5 


16 


3 




4 


31 


7 


-162 


3 


4 


2 


~ -76 


5 


1 


3 




4 


3 


7 





= 2. z = 



3 


4 


17 




5 


1 


16 




4 


3 


31 


-228 


3 


4 


2 


~ -76 


6 


1 


3 




4 


3 


7 





= 3. 



The values of x^ y^ and z may be written as determinants directly, 
without substituting the values of the elements. The denominator in 
each determinant should be written first, and then the numerator, 
according to the direction given above. 



Solve the following linear equations : 



3aj — 2y— 2; = 4, 
9. ^5aj-3y+ 2; = 10, 
2x + ^y-Sz = ll. 

2a; -4^ + 32 = 10, 
10. -l 3x+ y — 2z = 6y 
x + 3y— 2 = 20. 



6a; — 3y+ 2 

11. ^9x-\-2y-Sz 
x — 4,y — 5z 

x+ y+ z 

12. ^ 5x-^4:y-^Sz 
3a; 4-4^ — 32 



= 16, 
= 14, 
= 10. 

= 6, 
= 22, 
= 2. 



DETERMINANTS OF ANY ORDER. 

648. A determinant with n rows and n columns is called a 
determinant of the nth. order. It contains n' elements. 

A determinant of the nth order may be developed by taking 
the sum, with the proper sign (to be established later), of all 
the possible products of its w* elements that can be formed 
by taking as factors one element, and only one, from each 
row, and one, and only one, from each column. Each term 
of the developed determinant will contain n elements 

Take, for example, the determinant : 

This is a determinant of the fourth 
order ; and the sum of all possible prod- 
ucts of its 16 elements in sets of 4, one 
being taken from each row and one from each column, will be its de- 
velopment or value. 



ai 


61 


Cl 


di 


as 


&2 


Ca 


d2 


as 


bs 


Cs 


ds 


a4 


64 


Ca 


^4 



348 



ALGEBRA. 



t§649. 



649. The product of the elements in the principal diagonal 
from ai to ^4, which is afi^c^i, is taken as the leading term. 
Its subscripts are in natural order from left to right (1, 2, 3, 4), 
and it is regarded as positive. 

The signs of the other terms are determined by the number 
of interchanges required to bring the subscripts in each in their 
natural order. If the number of interchanges is even, the 
term is positive; if the number of interchanges is oddy the 
term is negative. 

Thus, in a2&iC4d8, to bring 1 to the first place, it must be interchanged 
with 2, and to bring 3 to the third place, it must be interchanged with 
4, making two interchanges, and hence the term is positive. But in 
ai&4C8f?2, the 2 must be interchanged with 3 and with 4, and 3 must be 
interchanged with 4, making three interchanges, and hence the term is 
negative. 

650. A determinant may be indicated by its leading term 
in vertical lines. Thus, | % 62 1 denotes a determinant of 
the second order ; | ai 62 C3 1, a determinant of the third 
order ; | Oi ^2 Ps ^4 1> 3, determinant of the fourth order ; and 
I tti &2 C3 c74 ••• r„ I, a determinant of the nth order. A deter- 
minant may also be expressed by the notation 2 ± before 
its leading term, as 2 ± a^b^. 

MINORS AND COFACTORS. 

651. If we strike out of a determinant any number of rows 
and the same number of columns, the remaining elements 
form a determinant called a minor. This minor, and the de- 
terminant formed by the elements common to the rows and 
columns stricken out, are called cofactors. 

Take, for example, the determinant (1), and strike out the 
first row and the first column as in (2). Then (3) is its first 
minor, and Oi is its cofactor. 



(1) 



tti hi Ci di 




CI2 bi Ci di 


(2) 


<tt &s Cs ds 


04 &4 C4 (I4 





fli &i Ci di 

c^ h% C2 d^ 

as &8 cs dg 

94 &4 C4 d4 



(8) 



&2 Cs ^2 




63 Cs dz 


(4) 


64 C4 d^ 






653.] 



DETERMINANTS. 



349 



If we strike out the first and second rows and the first and 



second columns, as in (4), then 



p8 ^ 
C4 ^4 



its cof actor, or corresponding minor. 



is the minor, and 






652. From the determinant (1) above, we can obtain the 
cofactors. 



(1) 




(2) 




(3) 




(4) 


b2 C2 di 




61 Ci di 




61 ci di 




61 Ci C?! 


hs cs ds 


02 


bz cz ds 


as 


bi C2 dz 


«4 


&2 ^2 di 


64 C4 d4 




64 C4 d4 




64 C4 di 




bz Cz dz 



ai 



It will be seen that (1) contains all the possible terms that 
contain ai ; (2), all the possible terms that contain Oj ; (3), all 
the possible terms that contain Og; and (4), all the possible 
terms that contain a^. But in each one of these minor deter- 
minants there are six terms, as shown in § 646, and hence 
there are 24 terms (6 x 4) in a determinant of the fourth 
order. 

In like manner it can be shown that there are 120 terms, (24 x 5) or 
(1x2x3x4x6), in a determinant of the fifth order ; and it may be 
shown that the number of terms in a determmaut of the nth order is the 
continued product of 1 x 2 x 3 x 4 ••• x n. 

653. It further appears from the aboye that the expressions 



Ol 



taken together, comprise all the terms in the original deter- 
minant (1) in § 651. 

It is evident that the first is positive, since, when each term of the minor 
is multiplied by ai, the order of the subscripts is not changed. The 
second is negative, since, when each term of the minor is multiplied 
by 02, one interchange is necessary to bring a^ to the second place, 
llie third is positive, since, when each term is multiplied by az, two 
interchanges are necessary to bring az to the thii*d place. The fourth 
is negative, since, when each term is multiplied by a^, three interchanges 
are necessary to bring a^ to the fourth place. 



bi C2 di 




61 Ci di 




61 ci d\ 




bi c\ d\ 


bz Cz dz 


, a2 


bz Cz dz 


» «8 


b^ C2 d^ 


, and ai 


b^ C2 d^ 


64 C4 d4 




&4 C4 di 




64 C4 di 




bz Cz dz 



850 



ALGEBRA. 



[§654. 





&2 Ci di 




61 Cl <?1 




61 Cl <Ji 




61 Cl di 


=ai 


bz Cs dz 


-aa 


&8 cs ds 


+ a3 


&2 Ca da 


-04 


6a Ca da 




64 C4 di 




64 C4 d4 




6$ Cs dg 




63 Cs ds 



Hence the following identity : 

ai 61 Cl di 
aa &a <Hi da 
as bz Cs dg 
a4 64 C4 d4 

It is evident from the foregoing conaiderations that a determinant of 
the nth order can be made to depend on n determinants of the order of 
n-1. 

PROPERTIES. 

654. Let us here recapitulate the properties of determinants 
already established; and add a few others which are of special 
value in manipulating determinants. 

1. The value of a determinajit is not cUtered by changing the 
roivs to columns, and the columns to rows. 

This has been shown to be true of determinants of the second and third 
orders (§§ 638, 643). It is also true of determinants of any order, since 
the succession of terms is the same in the changed form as in the original 
determinant. 

2. If any two rows or columns be interchanged, the sign of the 
determinant will be changed, but its value will not be altered. 

This has also been shown to be true of determinants of the second and 
third orders (§§639, 645). It is generally true; for, if the rows inter- 
changed be adjacent, the changed form will require one more interchange 
than the original form to bring the subscripts into regular order ; and, if 
the rows interchanged are not adjacent, an odd number of interchanges 
will be necessary. 

3. If two rows or columns of a determinant are identical, the 

determinant vanishes. 

For let D be the value of the determinant. Then, if the identical rows 
be interchanged, the sign of the determinant will be changed by Property 2, 
but its value will not be altered. Hence i>=—i>. .•. 2 2>=0. .*. Z>=0. 

4. If each element in any row or column be multiplied by the 
same number, the determinant will be multiplied by that number. 

This is evident from the fact that every term of a determinant contains 
one, and only one, element from the same row or column. 

mai &i 



Hence 



ma% b% 



= maibi — fna%bi = m(ai&a — a^i)* 



§654.] 



DETERMINANTS. 



S51 



5. If the elements of any row are like muUiplea of any other 
row, the determinant vanishes. 



For 



mai 


ai 


bi 




Ol 


ai 


bi 


mai 


a^ 


bi 


= w 


as 


as 


62 


tnaz 


as 


68 




as 


a8 


bs 



= 0. 



The determinant is put in the second form by Property 4, and this in 
turn vanishes by Property 3. 

6. A determinant of the nth order can he expressed in n deter- 
minants of the (n — l)th orderc 

This has been proved in § 653. 

7. If all the elements hut one of any row are zeros, the order of 
the determinant is reduced hy one. 

For if A\t Az, As, A4, etc., be the minors corresponding to au <hi as* 04, 
etc., and if D denotes the determinant, then 

D = aiAi — 02^2 + asAs — a^A^ + etc. 

Now, let Di represent the determinant when all the elements au (h^ 
Qif 04, etc., except one (say ai), are equal to zero, then Di = aiAi. But 
the order of 2>i, which is the same as that of Ai, is one less than D : 
hence the property is proved. 

8. If every element of any column or row is a sum of two or 
Tftore numbers, the determinant is equal to the sum of two 07' more 
determinants of the same order. 















ai + fc + Z 


6, 


Cl 














For 


02 + wi + n 62 C2 

08 + » + p 68 Cz 




= (ai+* + 


62+ C2 
68 Cz 


-(02 + m+ n) 


61 Cl 
68 C8 


+ (08 + « + p) 


61 
62 


= ai 


62 C2 

68 Cz 


— as 


61 Ci 

68 Cz 


+ 08 


61 Ci 

62 C2 


+ / 


62 C2 
68 C8 


— m 


61 Cl 
68 C8 


+ « 


61 Ci 

62 C2 


-\-l 


^2 C2 

bz Cz 


— n 


61 Ci 
68 Cz 


-'J 


1 Cl 

2 Cj 






ai 61 ci 




k 61 ci 




Z 61 ci 




=s 


02 ^2 ^2 


+ 


9n 62 C2 


+ 


n 62 C2 


 




a 


8 68 < 


58 




8 6 


B 


Cz 




P 


68 


cl 















Cl 
C2 



862 



ALGEBRA. 



[§655. 



9. The value of a determinant is not altered by adding to or 
subtracting from the elements of any column the corresponding 
elements of the other columns multiplied by the same factor. 



For 



052 + «i&2 + WC2 6a 



Cl 




a\hici 




6161C1 




C161C1 


ca 


= 


(Za^aCa 


+ w 


&262C2 


+ n 


C262C2 


C8 




azhzCz 




bzhzCz 




Cs^sCs 



But by Property 5 the lajst two determinants vanish, and hence 



fli + fiibi + nci 


61 


Cl 




Ol 


61 


Cl 


02 + m62 + wca 


&a 


ca 


= 


02 


62 


Ca 


as + wfes + ncs 


6» 


C8 




08 


68 


C8 



In like manner it may be shown that the value of a determinant is not 
altered by subtracting elements as stated in Property 9. 
This property holds true for a determinant of any order. 

10. If the signs of all the dements of any column or row be 
changed, the sign of the determinant will be changed. 
For this changes the signs of each term of the determinant. 



655. The application of one or more of the above properties 
will make apparent the changes in the following determinants : 



8 
3 
4 



1 
5 
9 



6 

7 
2 





15 1 6 




1 1 6 




= 


15 5 7 
15 9 2 


= 15 


15 7 
19 2 


= 15 



1 


1 


6 





4 


1 





8 


-4 





4 


1 


= 15 


8 


-4 



= 16(_ 16 - 8) = - 15 X 24 =- 360. 



The second determinant is obtained from the first by adding the 
second and third columns to the first, agreeable to Property 9. The third 
determinant is obtained from the second according to 4. The fourth is 
obtained from the third by subtracting the first row from the second and 
third rows respectively, according to 9. The fifth is obtained from the 
fourth according to 7. Then the determinant of the second order is 
developed. 

Trace the following changes in the same determinant : 



8 16 




3 5 7 


r= 


4 9 2 





15 15 


15 




1 1 1 




1 







2 
6 


4 
-2 


3 5 

4 9 


7 
2 


= 15 


3 5 7 

4 9 2 


= 15 


3 2 

4 5 


4 
-2 


= 15 



= X6(- 4 - 20) = - 15 X 24 =- 300. 



§ 656.] 



DETERMINANTS. 



858 



Trace also the following changes : 



8 16 




3 6 7 


= 


4 9 ,2 





8 1 6 




8 16 




15 16 16 


= 16 


1 1 1 


= 16 


4 9 2 




4 9 2 





7 




1 

1 



-6 9- 



=15 



7 
6 



6 

-7 



= 16(- 49 + 26) = - 15 X 24 = - 360. 



The foregoing determinant is a magic square, and can be reduced in 
many different ways. 

Trace the following changes in the magic square determinant 
of the fifth order : 



17 24 


1 

4 


I 8 16 








1 


1 1 


1 1 




1 








23 6 7 14 16 




23 5 7 14 16 




23 -18 -16 -9 -7 


4 6 13 20 22 


=66 


4 6 13 20 22 


=6. 


5 4 2 9 16 18 


10 12 19 21 3 




10 12 19 21 3 




10 2 9 11 -7 


11 18 26 2 9 




11 18 26 2 9 




11 7 14 _9 _2 




18 16 9 7 




-65 -90 70 




-65 -90 70 


= -66 


2 9 16 18 
2 9 11 -7 


= -66 


6 26 
2 9 11 -7 ' 


= -66 


6 25 
35 95 -46 




7 14 _9 _2 




1 _13 _42 19 




1-13 _42 19 




-13 -18 14 




1 20 -4 




1 20 -4 


=126x65 


1 6 


= 125x65 


15 


=125x66 


16 






7 ] 


L9 


— 


9 






7 19 


-9 






-121 19 



= 125 X 65 



1 5 
-121 19 



= 125 X 66(19 + 605) = 125 X 65 x 624 = 5070000. 



The pupil may find amusement, as well as practice, in reducing this 
determinant in a great many different ways. 



656. By applying the principles now established, we can 
readily solve groups of simultaneous linear equations contain- 
ing two, three, or more unknown numbers. 



Take, for ezr«mple, the equations solved in § 647 

flix + hiy -{■ ciz = di, 
a^x -\- h^y + c^z = (^, 

white's alo. — 23 



G) 

(2) 
(3) 



854 



ALGEBRA. 



[§ 657. 



62 C2 

63 Ca 
and add the results, we have 



If we multiply (1) by 



(2) by - 



61 Ci 
63 C3 



, and (3) by 



61 ci 

62 C2 



+ 






62 C2 

63 Cs 

62 C2 

^3 C3 



-a2 



-62 



61 Ci 

61 Ci 
63 Cs 



+ 08 



+ &8 



62 C2 



62 C2 

63 C8 


-C2 


&1 Ci 
63 Cs 


+ C8 


62 C2 

63 C3 


-d2 


63 Cs 


+ (?3 



62 



ci|\ 

C2 / 



= di 



But by Property 6 this may be written 



bi ci 
62 C2 



ai 


bi 


Cl 




a2 


&2 


C2 


« + 


as 


bz 


C3 





61 &1 Ci 




62 &2 C2 


y + 


63 63 C3 





Cl 


61 


Cl 




C2 


62 


C2 


« = 


C3 


bs 


C3 





d\ 61 Cl 

^2 &2 C2 

ds bz Cs 



Now, by Property 3 the coeflBcients of y and z vanish ; and hence 

di 61 Cl 
df2 62 C2 
ds 63 Cs 



a; = 



«i 61 Cl 
a2 62 C2 
as &8 Cs 



657. In like manner the values of y and z are obtained. 
In finding these values, instead of rearranging the terms, v^e 
change the signs of the determinants for y by interchanging 
the columns containing the (Vs and 6's ; and for z, by inter- 
changing the columns containing the d's and c's. 

Similarly, the values of the unknown numbers in a system of 
linear equations with any number of unknown numbers may be 
written down in determinant form. 



§ 660.] 



CUKVE TKACING. 



S55 



CHAPTER XXIII. 



CURVE TRACING. 



p' 



668. The position of any place on the earth's Surface is 
defined if its latitude and longitude are given, the latitude 
being measured north and south from the equator, and the 
longitude east and west from any convenient meridian. 
North latitudes are defined as positive^ and south latitudes 
as negative; east longitudes as positive^ and west longitudes 
as negative (§ 60). 

659. Similarly, we can locate the position of a point in a 
plane if we know its distances 

and directions from two fixed ^ 
intersecting straight lines. Let 
these lines be drawn perpen- 
dicular to each other, as XX' xi- 

and YY, and let them inter- 
sect at O. 

The line XX' is called the 
axis of X or the axis of abscis- 
sas; the line YY, the a^s of 
y or the axis of coordinates; and the two together, the axes of 
coordinates. The point is called the origin. Each of the 
four portions into which the two lines divide the plane of the 
paper is called a quadrant, and the four quadrants are num- 
bered as indicated in the figure. 

660. Distances measured along OX, or parallel to OX, from 
YY to the right, are defined as positive; along OX', or parallel 
to OX', from YY to the lefty as negaJtive. Distances measured 



1^ 



8 



O N 

4 . 
P 



/// 



IF 
Fig. 1. 



356 ALGfiBKA. [§ 661. 

along OF, or parallel to OF, from XX' upward, are defined as 
positive; along OF, or parallel to OY^, downward, as negative. 
Thus, if the point P is 3 units of length from YY^ in the posi- 
tive direction, and 2 units of length from XX' in the positive 
direction (some convenient unit of length being taken), its 
position is completely determined ; for from O we have only to 
measure along OX a distance ON equal to 3, and then from 
N measure upward and parallel to O F a distance NP equal 
to 2, and we arrive at the position of the point P. 

661. The two distances ON and NP are called the coordi- 
nates of the point P. Distances along the axis of x are usually 
designated by the letter x, and distances along the axis of y by 
the letter y. Then, in the example given for the coordinates 
of P, we have a; = 3, and y = 2, visually written (3, 2), or in 
general (x, y). The point (—3, 2) is found in the second 
quadrant by measuring from O along OX' a distance ON 
equal to — 3, and then from N measuring upward and parallel 
to F a distance N'P' equal to 2, and we arrive at the position 
of the point P'. 

Similarly the point P", whose coordinates are (— 3, — 2), is 
found in the third quadrant ; and the point P'", whose coordi- 
nates are (3, — 2), is found in the fourth quadrant. 

662. It is thus seen, that, to determine the position of a point 
in a plane, it is not sufficient to know simply the distances 
of the point from the two lines XX' and YY', but the direc- 
tions must also be known. If the distances alone were known, 
four points, one in each quadrant, would satisfy the conditions; 
but, when the directions also are assigned, the position of the 
point is completely determined. 

Exercises. 

1. What are the coordinates of the origin? 

2. In what quadrants are the following points: (2, —6), 
(-3,5),(-4, -1), (a,6)? 



§663.] CURVE TRACING. 357 

Draw the axes of coordinates and locate these points, using 
J of an inch as a unit of measure ; and use the same unit of 
measure in each of the following examples. 

3. Locate the following points : (7, 8), (1, 0), (0, 3), (0, 0), 
(5, - 4), (- 1, - 7), and (- 5, 0). 

4. Locate the following points : (— 3, 2), (— 3, —2), (3, —2), 
(0, 3), (- 1, 2), (1, 2i), (i, - 2). 

5. Draw the triangle which has for vertices the following 
points : (1, 1), (- 3, 2), (0, 0). 

6. Draw the quadrilateral having for its vertices the follow- 
ing points : (2, - 3), (- 3, 4), (3, - 4), (- 2, 3). 

7. Draw the polygon having for its vertices the following 
points: (2, 0), (1, 1), (- 1, 1), (0, - 2), (- 1, - 1), (1, - 1). 

The student is recommended either to provide himself with paper 
ruled in small squares, or to draw on the paper two straight lines perpen- 
dicular to each other, and lay off distances from the origin along these 
lines equal to ^ of an inch, and draw lines parallel to the original lines 
from these points, thus dividing the paper into small squares. 



GEOMETRICAL REPRESENTATION OF EQUATIONS WITH 
TWO UNKNOWN QUANTITIES, X AND F. 

I. Equations of the first degree in x and/. 

663. Let 2x-\-y — l=:Ohe an equation of the first degree 
in X and y. For every value of x we have a corresponding 
value of 2^ ; so that, if x varies continuously, y also varies con- 
tinuously. 

Let a number of corresponding values of x and y be found. 
Thus, 



If a; = 0, 


y = l. 


(2) 


If «=-l. 


y= 3. 


If a = 1, 


y = -l. 




If a; = -2, 


2^= 6. 


If x = 2, 


y = -3. 




If «=-3, 


y= 7. 


If a; = 3, 


y = —6. 




If a; = - 4, 


y= 9, 


I£x = 4, 


y=-7. 




If »=:-6, 


y = ll. 



868 



ALGEBRA. 



[§ 664, 



These values of x and y form the co5rdiiiates of a set of 
points which can readily be located. When these points are 
connected by a line, we have a geometrical representation or 
picture of the equation. This line is called the locus of the 
equation ; and hence the locus of an equation may be defined 
as the line the coordinates of every point of which satisfy the 
given equation. 

If all possible values of x should be substituted in the equa- 
tion, and the corresponding values of y found, the sets of 
values would represent the coordinates of an infinite number 
of points ; and these points, if located exactly with reference 
to the axes of coordinates, would form a continuous line or 
locus. In practice, only those values of x are taken which 
are most convenient, usually integral, and the coordinates of a 
set of these points found. 



664. To construct the locus of the equation 2aj-f-y — 1 = 0, 
let us take the first four positive values of x given in § 663. 

Draw the axes of coordination, and 
assuming some convenient unit of meas- 
ure, say ^ of an inch, mark the paper 
into squares, as in the figure. 

The point (0, 1) is found at a, a dis- 
tance of 1 unit above O. The point 
(1,-1) is found by measuring a dis- 
tance equal to 1 on OX, and then a 
distance equal to 1 below OX, parallel to 
or, the point 6. The point (2, -3) 
is at c, whose distance from O F is 2, 
and from OX is — 3. The point (3, —6) 
is at d, whose distance from OY is 3, 
and from OX is — 6. Through the points a, 5, c, and d draw the indefi- 
nite line AB, and we have the geometrical representation or locus of the 
equation 2a; + j/ — 1=0. 

The foregoing process might be continued indefinitely for both posi- 
tive and negative values of x. Fractional values might also be used, and 
points located between those found above. 



\^ r 


^ 


^ 


_5 


- I- K 


y 


X' «v x- 


^h 


\ 


\ 


'S^ 


\ 


"^^ 


^ 


\ 


F' Ml 



Fig. 2. 



$685.] CURVE TRACING. 869 

EXBRCISBS. 

1. Construct the locua of 2x + y = 6. 
ScQGBBTTOM. Computc & table of coire- 

aponding values of x and y, and, nith these 
sets of corresponding valoea aa coordinates of 
points, proceed as above. The line AB in 
Fig. 3 is the locus of the equation. 

2. Construct the locus of y = 5x+3. 
The line Mlf in Fig. 3 is the required locus. 

3. Construct the locus of 

3x + 5y = 15. 

Let X and y in turn ^nal 0, giving x = 6, 
and y = S. 

Locate the points (0, 3) and (6, 0), and fiu, 3. 

through these points draw an indefinite line. 

Construct the loci of the following equations : 

4. 2a;+y = 6. 9. x — y = 0. 

5. 7x-iy = 0. 10. y = -ix + 6. 

6. iC + 5y = S. 11- 2x-^l=y. 

7. y=2x + 2. 12- ^~2 ^ ^^^ 

8. x-y = 3. 13. l + l = i- 

665. Each of the above loci will be found to be a straight 
line, and it may be assumed that every equation of the first 
degree in x and y is represented by a straight line; and, since 
two straight lines can intersect in but one point, the point of 
intersection of two straight lines is the geometrical solution of 
the two equations of these lines. 
Let us, for example, find the geometrical solution of the 
<x + y + l = 0, 
lx + 2y + i = 0. 



860 



ALGEBRA. 



[§ 666. 



Compute, as before, sets of values of x and y for each equar 
tion, and construct loci, as in Fig. 4. 

The lines will be found to inter- 
sect in the point P, whose coordi- 
nates are x = 2, and y = — 3. 

These values of x and y must 
satisfy both equations: for, since 
every point in AB must satisfy 
x-\-y+l=0, the coordinates of P 
must satisfy this equation. Like- 
wise, since the coordinate of every 
point in CD must satisfy 

x + 2y + 4: = 0, 

the coordinSites of P must satisfy 
this equation. Hence «=2, and y= —3, satisfy both equations 
simultaneously, and represent the solution of the equations. 

14. Find the coordinates of the point of intersection of the 
loci given in Fig. 3. 



















y 
















— 


B 






























\ 


































\ 


































\ 


























•V 


n 






\ 
































\ 
























i 


"V 


•*> 






\ 




















X 








^ 


s. 




\ 
















X 














"^ 


\ 


\ 
































^ 


K 


(2 


-s 


) 


























p 


'^ 


K 
































\ 


"V. 


V 




Q. 


























\ 




V 




























\^ 






















Y. 


f 








^ 







Fig. 4. 



Solve geometrically the following sets of equations : 

(2a?-f 32/ = 2, 
(6a; + 3y = 4. 



15. 



16. 



17. 



I aj + 2^ = 4, 
\x — y — (^, 

ix + 2y = lS, 
(3x + y = U. 



18. 



19. 



20. 



= -1, 



(3a; + 42/ = 
X9x-2y=^ 4. 

(4a;-f 5t/ = 22, 
\5x-2y = ll. 



II. Equations of the second degree in x and/. 

666. The points whose coordinates satisfy the equation 
y = x^ — 2x-\-2 do not lie in a straight line. They do, how- 
ever, lie in some line which is determined by the equation. 
To construct this line it is necessary to find a number of points, 
and then draw a continuous curve through them. Thus, 



§ 666.] 



CURVE TRACING. 



861 



For x = 0, 


y = 2. 


For x = l, 


y=l. 


For x = 2, 


y = 2. 


For a; = 2|, 


y = 3i. 


For a; = 3, 


y = 6. 


For X = 3i, 


y = 7i. 


For a = - J, 


y = 3|. 


For a; = — 1, 


y = 6. 



















Y 








— 




















































\ 










/ 






















\ 










/ 
































/ 
























\ 






/ 


























\ 






/ 


























\ 




/ 




























\ 


y 














X 






























X 



















































































































































































r 

















Fig. 5. 



The points tabulated above are plotted in Fig. 5, and a con- 
tinuous curve drawn through them. This curve is the locus 
of the equation y = ar^ — 2a? + 2. 

21. Construct the loci of y^ = 4aj + 2 (Fig. 6), and xy = 4: 
(Fig. 7). 



Writing the second equation 
in the form y = -, and assign- 

X 

Ing both positive and negative 
values to a, we get the follow- 
ing table of values for x and y. 



For a; = 0, 


y = 00. 


For x = ±l, 


y = ±4. 


For x = ±2, 


y = ±2. 


For a; = ± 3, 


y = ±t 

^ 3 


For a; = ± 4, 


* * . • 



















Y 












































































1* 


























V 




^ 


























^ 


,^ 


























/ 


^' 




























J 


/ 
















X 














( 
















X 
















\ 

































\ 


s. 
































\ 


\, 


































\ 


'^.. 
































^•5; 




^.. 


































^ 




















































r 

















For » = 0, 



y = oo< 



Fie. & 



862 



ALGEBRA. 



[§667. 



"" Y 










T 




i 




n 




^ 




^ 




^^ 


X\ 


^""^^ 


.^x^ 


X 


1 '"'^1 




4 K 




> 




" -"-4-'T 








r 




"::.:.._... t 




1 


Y^ 



Fig. 7. 



Construct the following loci ; 

22. 4aj = 2^. 

23. 3aj2-4y2=12. 

24. 3aj2 + 42/2 = 12. 



Plotting the points that corre- 
spond to the positive values of x 
and y, we have a branch of the 
curve in the first quadrant. 

Corresponding to the negative 
values, we have a branch of the 
curve in the third quadrant. 

It is thus seen that in this case 
the curve is composed of two 
branches, as indicated in Fig. 7. 



26. aj2 + 2^ = 4. 

26. a^ = 4:y, 

27. a; + y + »* = — 1. 



GEOMETRICAL REPRESENTATION OF THE ROOTS OF 

AN EQUATION. 

667. Let t/ = a^+2a; — 4. Compute 
a table of values of x and y and con- 
struct the locus (Fig. 8). At the 
points Pi and Pg? where the curve cuts 
the axis of a?, we have y=0, and hence 
ic^4-2aj — 4 = 0. The values of x for 
these points are OPi and OP2; and as 
these are the values of x which make 
the expression a^-f2a; — 4 = 0, they 
represent the values of the roots of 
the equation. These are readily seen 
to be — 1 + V5 and — 1 —VS. 

668. If the value of the absolute 
term be increased, the curve will be 





I 








Y 










\ 




















\ 














/ 






\ 


















\ 














1 






\l 




















- -■ 






1 










i 








2\ 


-  





 -■ 


— 


J 






x; 















1 




X 






(, 






i 


— 








\ 












\ 






t 














\ 


/ 






















r 









Fio. 8. 



§689.] CURVE TRACING. 36» 

moved upward, and the points P, and Pj will approach each 
other. When we have y = a^+ 2 a? + 1, the points Pi and Pj 
coincide, and the curve simply touches the axis of x, or, we 
might say, it cuts the axis of x in coincident points. For these 
points we have a5* + 2a? + l = 0, the roots of which are readily 
seen to be each equal to unity. This is the geometrical repre- 
sentation of the condition of equal roots. Thus, when the 
curve touches the axis of a?, but does not cross it, the two values 
of X are equal. 

669. If the value of the absolute term be still further in- 
creased, the curve is moved upward still farther, and does not 
cut the axis of x at all, or it is said to cut the axis in imaginary 
points. In this case the roots are imaginary. Thus, if in ^ = 
0^ + 2 a; + 2, we put y=^0, and find the corresponding values 
of X, we get X = — 1 + V— 1 and — 1 — V— 1. These values 
of X aye imaginary, and the corresponding points cannot be 
located by means of the system of coordinates we are using. 
HencCj^ when the locus does not cut the axis of Xy the roots of 
the equation resulting by putting y = are imaginary. 

Construct the loci of the following equations, and determine 
the values of x which make 2^ = : 



1. 


y = iC^ - 1. 


6. 


y = oi^ — 2x. 


2. 


y = x^-\'l. 


7. 


y = 2ix^ + i. 


3. 


4y = ^-2aj2 + 16. 


8. 


y = ± 2 Va? -h 4. 


4. 


y^ = 2-x^. 


9. 


f = 16-a?. 


5. 


43^ = a^. 


10. 


^ =z= a^ — 6 a?^ + 11 a? — 6. 



864 ALGEBRA. [§ 670 



CHAPTER XXIV. 

PERMUTATIONS AND COMBINATIONS. 

PERMUTATIONS. 

670. The different orders in which a set of things can be 
arranged are called permutations. 

Thus, the permutations of the three letters a, 5, c, taken 

two at a time, are 

ab, hay ac, ca, be, cb. 

In like manner the three digits 1, 2, 3, taken two at a time, 
express the six numbers 

12, 21, 13, 31, 23, 32. 

671. The number of permutations of n things taken m at a 
time is equal to the eontinued product of the m successive integers 
from n to n — (m — 1). 

For suppose the n things to be n letters, a, b, c, €?,•••, and 
denote by "pi, "pj? "Ps? ••• "^Pn^ the number of permutations that 
can be formed from n things taken 1, 2, 3, ••• m at a time. 
The number of permutations of n letters taken one at a time 
is evidently equal to the number of letters ; that is, 

"p, = n. 

If we put each of the n letters before each of the remaining 
n — 1 letters, we obtain 

"i^a = "Pi X (n — 1) = 71 (n — 1). 

Again, if we put each of these **p2 permutations before each of 
the remaining n — 2 letters^ we obtain 

•i>3 = *i>8 X (n — 2) = n (n — l)(n — 2). 



§ 674.] PERMUTATIONS. 366 

Again, if we put each of these "pg permutations before each of 
the remaining w — 3 letters, we obtain 

"1>4 = "Ps X (n - 3) = 7i (n - l)(n - 2)(n - 3). 
We may proceed in this way until we obtain "p«_i; and, if 
we put each of these "p«_i permutations before each of the 
remaining n — (m — 1) letters, we shall obtain 

"P«="P»-iX[n-(m-l)]=w(n-l)(n-2)(n-3)...[7i-(m-l)]. 

672. If all of the n letters are to be arranged together, then 
m in the above formula becomes equal to w, and w — (m — 1) = 
71 — w -f 1 = 1, and 

»p„ = w(n - l)(?i - 2) ... 3, 2, 1. 

Hence the number of permutations of n things taken all 
together is equoU to the continued product of all the n integers 
from n doum to 1 or from 1 to n. 

Thus the number of permutations of four digits taken 
together is 4 . 3 • 2 . 1 = 24. 

673. The continued product of the successive numbers from 
n to 1 or from 1 to n is denoted by the symbol [w, which is 
read " factorial ti." Thus [5^ denotes 5.4.3.2.1. 

The symbol n! is also used to express factorial n. 

674. In the above formula for "p„, all the letters were sup- 
posed to be different. But if we suppose p of them, for exam- 
ple, to be the same, then the \p permutations, arising from the 
presence of these letters when different, reduce to 1. If we 
suppose another group of q letters to be the same, then \q 
permutations, in like manner, reduce to 1. 

Hence, to find the number of permutations of n things taken 
all together, one set of p of them being alike and another set 
of q of them being alike, we must divide ^p^ by \p and \q; 
and the required number of permutations will then be expressed 

by 1 , ; and similarly if there were three or more sets of like 
things. 



366 ALGEBRA. [§ 674. 



EXERCISBS. 

1. How many numbers can be expressed by 9 digits taken 
4 at a time ? 

Here n = 9, and m = 4, and hence 

9p4 = 9-8 -7 .6 = 3024. 

The subscript 4 shows how many factors are to be taken, the super- 
script 9 being the first. 

2. In how many different orders can 6 boys sit on a bench ? 

% = 6. 5. 4. 3. 2.1 = 720. 

3. How many permutations can be formed of the letters in 
the word possessions, taken all together ? 

Here n = 11, and the letter s occurs 5 times, and o twice. 

"»« Hi 11 . 10 . 9 ... 3 . 2 • 1 
»«"''« [^ = (Ig= 6.4.3.2. 1.2.1 = ^««^^- 

4. In how many ways can the letters a, 6, c, d, e, be 
arranged, taken 3 at a time ? Taken 4 at a time ? 

5. In how many ways could the seven prismatic colors be 
arranged, taken all together ? 

6. How many signals could be made with 5 flags of differ- 
ent colors, taken 1, 2, 3, 4, and 5 at a time ? 

7. In how many ways could a party of 5 persons be seated 
at a table ? 

8. How many permutations could be formed from the let- 
ters of the word Columbia, taken all together ? Taken 5 at a 
time ? 

9. How many signals in all could be made with 7 flags of 
different colors ? 

10. How many permutations can be formed from the letters 
of the word consonant, taken together ? From the letters of 
the word Mississippi f 



§ 678.] COMBINATIONS. 867 

COMBINATIONS. 

675. The different sets that can be formed from a given 
number of things without regard to the order in which they 
occur, are called combinations. 

Thus, the possible combinations of the letters a, &, c, taken 
two at a time, are db, ac, be, since db and ba, though different 
permutations, are the same combination, as are also ac, ca, and 
be, cb. 

676. Any combination of m things will produce [m permu- 
tations, for the set of m things that form a given combination 
can be permuted in [m different ways. 

677. The number of combinations of n things taken m at a 
time is equal to the number of permutations of n things taken m 
at a time, divided by [m. 

For if we denote the number of possible combinations of n 
things taken m at a time by "(7„, then, by § 674, 

... ng np ,^^n(n-l)(n^2).-.[n~(m--l)1 

" — [m 

678. Since for every combination of m things which we take 
out of n things we leave a combination of n — m things, it 
follows that the number of combinations of n things taken m at 
a time is equal to the number of combinations taken n — m at a 
time ; that is, "Cm = °Cn_ni. 

Exercises. 

1. How many combinations can be formed with the letters 
a, b, c, d, e, taken 3 at a time ? 
Here n = 6, and m = 3 ; and hence 



868 ALGEBRA. [§ 678. 

The number of combinations of 6 letters taken 2 at a time is also 10, 

since ^-^ = 10. 
1x2 

2. How many different pickets of 5 men and an officer can 
be made from a squad of 20 men and 3 officers ? 

Here n = 20, and m = 6 ; and hence 

^G, =20.19.18.17.16 ^ ^^^^ 
1.2.3.4.6 

Since one of 3 officers may go with each picket, the whole number of 
pickets is 16504 x 3 = 46612. 

3. How many combinations can be formed of the letters of 
the word longitude, taken 4 at a time ? How many combina- 
tions taken 5 at a time ? 

4. How many sums can be formed of the digital numbers 
1, 2, 3, 4, 5, 6, 7, 8, 9, taken 3 at a time ? How many taken 
6 at a time ? 

5. A number is the product of 5 different prime factors, 
including unity. How many divisors has the number ? 

6. How many different selections of 4 pieces each can be 
made from 12 different pieces of money ? 

7. How many different sums of money can be formed of 
a cent, a three-cent piece, a half dime, a dime, a quarter, and 
a half dollar ? 

8. In a school of 18 boys and 15 girls, how many classes 
could be formed, each to consist of 4 boys and 3 girls ? 

9. Out of the 7 prismatic colors how many combinations 
of 3 colors each can be made ? 

10. Company A contains one man more than Company B, 
and the number of combinations of 3 men each that can be 
made from Company A is to the number of similar combina- 
tions that can be made from Company B as 21 to 20. How 
many men in each company ? 



APPENDIX. 



-•o^ 



679. The following supplementary exercises are designed 
for advanced classes, and especially for students who are pre- 
paring to enter higher institutions which may have exceptional 
entrance requirements in algebra. Many of the problems have 
been selected from recent college examination papers; and 
each series contains as difficult problems as any found in the 
papers examined. 



680. Complex Fractions. (§ 273.) 
Simplify 

1 — a^ 1 — X 



1. 



2. 



3. 



1+a^ 1 + a? 

1—05^ 1 —X 

1 — OJ^/ X 



l-\-y\l-^x 



-1 



1-1 



1 a^^y^—x-{-y" 



1- 



24-1 
X 

X 



1- 



f 



'A 

1 

2 



X 



white's alg. — 24 



4. 



5. 



6. 



7. 



369 



faJ^ + ll 



x' + l 

. 03-1-1 > 
1 



a-\- 



[l-^x(x-l)l 



' + 0^ 



"-l 



y a? 

if 



*(- 



x + y I iB« + .v« 

x — y Qi^ — 'tf 

g; — y a^ — y^ 

x + y a^ + f 



370 ALGEBRA. [§681. 

a + b \ a + bj (3^-f)(x+y) 

1 x + y 1 a; — y ar'+(& + c)!g+6c a^ — cP 

^' 1 y + x '^1 y — x ' ' iii? + {b + d)x + bd 

y 1^ + 0? y y' + a? ti?+(a — c)x — ac 

681. RaUonaUzing the Divisor. (§ 398.) 

J VlO ^ (3+V3)(3+Vg)(v^-2) 

V5+V2-V6' ' (5-V5)(l+V3) 

2. 4+V2-V3 g __1 

* 3-V2+V3  l+V-2+V^3 

3. 1+V3+V6 . g V3 2-V^2 
1+V2-V3 • 2_V3 2+V=^' 

4 V(3 + 2V2)-V2 J 

• V(3 - 2 V2) + V2' 10. ^25-^20 + 2^^- 

g^ ^ + V 6 ^ SuGGBSTiON. — Multiply both 

5V3-2V2-V32+V50* terms by \/6 + v^. 

. 3 + 2V2-3V3-2V6 ,- 1 

6. — • 11. 



1 _|. V2 - V3 - V6 V25+ VlO +V4 



12. (Jl±^^^l^\J ^'-^' . 
\\a-b yia-\-bJ\ (a-\-by - ab 

682. Equations involving Radicals. (§ 410.) 
Solve 
1. Va4-V4a+ic=2V6-f-a;. 3. Va—x-\--\/b—x= 



^b^ 



X 



2. ^ 4, ^ ^ J^ . 4. V2+a;+^-fa?4-V2-a; 
' Vic-l Vic+l Va^-1 =V2-ic. 



§ 684.] 



APPENDIX. 



371 



5. 



3* 



6. 



V5a + 2 3V5^-8 



7. 



8. 



9. 



10. 



Suggestion. — Equations 5 to 10 may be solved by proportion. 

11. 



Va + Vci — X 1 
Va — Va — X ^ 

3^+ vr^_a+ VF^^ 



12. 



-\/x-\-a-\-Vx—a _ a-\-b 13. 
Vif+a— Va;— a a— 6 
V4a;4-1 +V4^ _-|^j 
V4iB + l— V4« 14. 



Vl4-a;+Vl4 + a; = 4- 
f •y/2x — y = Vx — y + 1, 

■y/x -y/y 
' +-4 = 4. 



VS V3/ 

1 _1 

x" — 6a5 " = 1. 



683. Imaginary Numbers. (§ 420.) Simplify 

1. (6-5V^)(5 +3V^). 5. - 8 V26 -5- 2 V^:^. 

2. (2.fV^r6)(3 4-2V^=TI)). 6. (3-V^l 

3. v^=nro-T--V6. 

4. Vl5-5-V^^. 
9. 



7. (3-V-5)- 



8. (3-V-l)-5-(2 + V 
(1+V^=^+V^(1+V^-V 



-1). 



12. 



3). 

10. (9V^-iiV^^)^(V^+V^=^). 

11. (-2V6-V^H-(V^=^-V2+V^^). 

14. V40 - 42 V^. 

15. V-5 + 12V^=l. 



a-^b^Z—l a — bV^^ 
a — bV^^ a + ftV— 1 



13. V34 - 12 V^. (§407.) 16. V_26 4-6V^=^. 
684. Fractional and Negative Exponents. (§ 436.) 
Extract the square root of 

1. a^y"^ — 4 x'^y'^ + 6 — 4 x'^y^ + x'^y^. 

2. a^ — a*VS + 2a*V^ + 4ic(aj — f|a). 
Extract the cube root of 

3. 8 «-* - 12 aj"'V- 4- 6 ic"* - a?"*. 

4. ai-9a* + 33a^-63a + 66a*-36a* + 8a*. 



872 



ALGEBRA. 



[§ 685. 



Resolve into factors 
6. x — y. 7. a"^ — &. 

6. aj*"+^ — x~l 

Simplify 



8. x^ — y\ 



9. x — x^y^-{-\y, 

3 3 X 

10. a;"2 --2a;"^2/*+y. 



11. 



12. 



J 



a^a ^. 



18. ^^^ r X 



aj-^y 



aJ^2^ 



-» 



a 



3e+jf\ X 



a" \'-* 



a? 



a» 



' — X 



y otyiw-ab- 



13. [(a;^"/*)^ -«*]«• 



19. a^6 ^ X a h ^ x a^6' 



20. (a*6~^)* X (a-«6~^)i 



-2/. 2 



14. 



a-26 



X 



a'-hb' 



3a 



3a 



a-^ + b ''" a ' 



21. x^y ^XX "225-* x(2/22J*)^- 



( _i 1 1 -i^V"^'\5 

15. \a ^x^yax 3 Vic^y . 
9"^* X ^3"+^ 



ah 3a 

ya5\ 2 n -2J 



16. 



17. 



3V3-" 
9-2 ^ 16* 



81"'^ ^ 4* 



26. 



e- 



4-e 



-i\/-r 



+ 



22. [(x^y ^^yy. 

23. (a;^»-*^* -^ aj^"+'>*) x a?*«». 

1 

24. [(«"•-")'"+'* X (a-"*)"-"]*. 

25. (e' + e-*)*-(e'-0'- 



rV^-1 



-xver>« 



2 ; • V 2V-1 

27. ic^+^ft^^-' X af-Y"+^ -!- m^+^y^, 

28. f (» - 1)* (aj + 1)"^ + i (^ 1)^ (« - 1)"*. 

29. (a6-^ + l)[a-i(l+V^3)6]x[a-i(l-V^3)6]. 



685. Quadratic Equations. (§§475-485.) 
Solve the following groups of equations : 
J [0^ + 2/^ = 189, 

V2/^ + 2/' = 10, 
icy* + 2/ = 4. 
(x' + xy-\-4:f = e, 
' [Sx' + Sf^U. 



(x'+f--l5{x-hy)=-My 

*• l2a:2/4-31(aj + 2/) = 289. 



2. 



5. 



6. 



X" + 2/** = ct, 
xy=b, 

ra^ + 2/* = 17, 



^ a? + y = a?2^ + 1. 



§ 688.] APPENDIX. 873 

686. Equations of Higher Degrees. (§§ 471, 472.) 

1. 3aj8 + 7ar'-h3aj + 2 = 0. 

Suggestion. — Divide by x + 2, and then solve the resulting quadratic. 

2. 64a^-39iB*-26aj4-16 = 0. 

3. i»* + 6aj3 + 5ic2 --12a;-12 = 0. 

4. afi + 15a3^-j-Ua^ = 0. 

5. aj*-10aj8 + 35iB2-50a; + 24 = 0. 

6. aj*-a?*-64a5* + 64a? = 0. 

7. a^-2aj8 4-0.-132 = 0. 

687. Proportion and Progression. 

1. If a:b = c:d, prove a:b =\^3a? -\- 5c^:-VSb^ -^ 5d\ 

2. Find a mean proportional between (1) 4a^ — 3« — 1 
and x-l; (2) 2aj-3 and 2a^4-aJ^-4aj- 3. 

3. Two numbers whose difference is d, are to each other as 
a to b. What are the numbers ? 

4. Prove that in a geometrical progression s = ^ ~^ ' 

?• — 1 

5. Show that if a -f 6, 26, 6 + c are in harmonic progres- 
sion, then a, 6, c will be in geometrical progression. 

6. The sum of the first six terms of a geometrical pro- 
gression is 728, and the sum of the third and fourth terms 
is 72. Write the series. 

688. Convergent Infinite Series. (§ 573.) 

7. Show that in a decreasing geometrical series to infinity 

a 
s = 

1-r 
Find the sum to infinity of 

«• -8-1-^--. 12. 1+A. + U.... 

10 1 -1 i - _^_ 



874 



ALGEBRA. 



[§ 689. 



Find the value of the following recurring decimals : 

14. .24. 15. .851. 16. 62l. 17. .037. 18. .12135. 
19. Show that 0-^-a=0; a-5-0=oo; and 0-5-0= any number. 



689. Exponential Equations. (§ 613.) 
Solve the following exponential equations : 

1. 23'+« = 5*-^ 3. 7*' = 2»+*.3*. 

2. 3^-^ = 2.5'+^ 



4. ^1""+* = %. 



5. 3*-*-^' = 4-^ 

6. 2*"-i = 3^-». 



690. Undetennined Coefficients. 

Resolve into partial fractions 



1. 



2. 



3. 



(x - S){x -h 1)* 
3a^^_3a; + 18 



«»-8 

Expand into series 
1-x 



4. 



5. 



6. 



lH-2a; 

a^(x - l)(x + 2y 

2a^H-^l . 
2aj*-f a;-3* 



7. 



1 -f-aj-hoj* 



8. 



1-f aj-ho* 

> . 

x — a^ 



9. 



l4-2a^-|-a^ 



691 . Determinants . 

Solve the following linear equations : 



1. 



2. 



3. 



4. 



5. 



6. 



a;-2y = 3, 
2x-{-5y = 5. 

r3aj-|-42/ = 4, 
6x-\-6y = l, 

6x — 5y = 7, 
7a; -22/ = 12. 

Expand the following determinants : 



'8aj + iy = 26, 
4a? — 5y = — 8. 

Sx + 5y = a, 
4aj4-2y = &. 

aa; -h &y = c> 
ca; -}- dy = a. 



7. 



a;-f 2/ a;— 2/ 



8. 



aJ-h2/ ar^+2/^ 

05— y «^— y* 



9. 



a;-2/ x^—xy-\-y^ 



ANSWERS. 



Note. — When several examples are given under the same number, in 
most cases the answer to the last only is given. Answerswhich, ii' given, 
would destroy the utility of examples, are omitted. 



Page 11.— 2. Son, $14; father, $42. 3. Vest, $4; coat, $20. 

4. Less, 7 ; greater, 35. 5. Less, 9 ; greater, 27. 6. Shorter, 5 yds.; longer, 
25 yds. 

Page 12. — 7. B, 50 sheep ; A, 100 sheep. 8. Less, 18 ; greater, 72. 
9. 20. 11. Father's age, 46 yrs.; son's, 15 yrs. 12. 135. 13. Saddle, 
$6; harness, $18. 

Page 13. — 16. Shorter, 45 ft. ; longer, 76 ft. 16. 50 and 70. 17. 1st 
person's, $576; 2d person's, $1226. 18. Chain, $20; watch, $66. 
19. 1st party, 158 votes; 2d party, 206 votes. 20. Men, 47; women, 
62 ; children, 218. 21. A, $3000 ; B, $ 1500 ; C, $7500. 22. 1st, 7 yds. ; 
2d, 21 yds.; 3d, 14 yds. 23. Shorter, 15 ft.; longer, 30 ft. 24. A's 
share, $1200; B's, $2400; C's, $4800. 25. A's share, $4; B's, $2; 
C's, $ 12. 26. First, $400 ; second, $600 ; third, $800. 

Page 14. — 27. A's age, 40 yrs. ; B's, 20 yrs. ; C's, 35 yrs. 28. 16. 
29. One, $60; other, $80. 30. B's age, 14 yrs. ; A's, 56 yrs. 31. Boys, 
80 ; girls, 160. 32. Buggy, $ 60 ; horse, $ 120. 33. Mother's age, 46 yrs. ; 
daughter's, 15 yrs. 34. A's share, $86; B's, $40. 36. 30 and 46. 
86. Smaller, 79 acres ; larger, 121 acres. 37. 186 sheep. 38. Standing, 
35 ft. 'y broken off, 56 ft. 39. B, $2000; A, $3500. 

Page 15. — 40. Each son, $3675 ; widow, $ 14,700. 41. Shorter, 24 in.; 
longer, 40 in. 42. 18. 43. 1st, 12 ; 2d, 24 ; 3d, 40. 44. Father's age, 
48 yrs. ; son's, 24 yrs. 45. Mother's age, 36 yrs. ; daughter's, 16 yrs. 
46. A's age, 72 yrs. ; B's. 36 yrs. ; C's, 18 yrs. 47. Father, $80 ; elder son, 
$ 40 ; younger, $ 20. 48. 1st candidate, 137 votes ; 2d, 97 votes ; 3d, 72 votes. 

Page24. — 1. 16. 2.18. 3.6. 4.31. 5.18. 6.26. 7.72. 8.20. 

Page 25.-9. 9. 10. 51. 11. 31. 12. 28. 13. y. 14. 2x-|^. 
15. 2 y, 16. 4. 17. - a. 18. ab -^ h. 19. 6-4, 20. 4 a + a6 + 2 6. 
21.2 6. 22. 2a + 7 6. 23. -2bK 24. -b^. 25. -2xy. 26. a;y-2y«. 

Page26. — 1. 10. 2.26. 3.7. 4.24. 5.51. 

Page 27. —6. 6 a. 7. 6 x. 8. 64 x - 18. 9. ^xy - 13. 

Page 29. — 1. 23. 2. 56. 3. 4. 4. 45. 5. 21. 6. 4. 7. 38. 
8. 0. 9. - 1. 10. 88. 11. 37. 12. 14. 18. 62. 14. 18. 15. 28. 

375 



376 ANSWERS. 

16. 35. 17. 92. 16. 0. 19. 7. 90. 22. 81. 11. 98. 169. 83. 28. 
94. 188. 85. 7. 

Page 39. — 1. z = 3. 8. z = 3. 8. z = G. 4. z = 8. 6. z = 8. 
6. z = 7. 7. z = 4. 9. z = 2. 9. z = 6. 10. z = 3. 

Page 40.— 11. z = 2. 12. z = 7. 13. z = 6. 14. z = 5. 

15. z = 11. 16. z = 7. 17. z = 3. 18. z = 10. 19. ^ = 10. 20. z = 3. 
81. Com, 120 bu. ; wheat, 240 bu. ; oats, 360 bu. 82. 1st, $ 14 ; 2d, 
$28 ; 3d, $84. 83. Quarters, 4 ; dimes, 16 ; nickels, 32. 84. 1st year, 
6900; 2d year, $1800; 3d year, $3600. 85. B*s, 62000; A*s, 64000; 
C's, 6 6000. 86. 9 and 27. 87. A*s, 6 28 ; B's, 6 7. 28. Younger, 8 y rs. ; 
elder, 24 yrs. ; father, 48 yrs. 

Page 41. — SO. In quarters, 63; in dimes, 6120; in nickels, 6-60. 
81. Younger, 6 18 ; elder, 636 ; father, 6 108. 38. A, 30 yrs. ; B, 40 yrs. ; 
C, 60 yrs. 38. 1st farm, 100 acres ; 2 J farm, 80 acres. 34. Each sister, 
170 acres; brother, 300 acres. 35. A, 6^0; B, 6130. 36. B, 62420; 
A, 64840 ; C, 6^760. 37. 9 hours; one travels 27 miles, other 36 miles. 

Page 42. — 38. First, 50; second, 80; third, 65. 39. Serge, 75^ per 
yiurd; silk, 61.50. 40. Oranges, 4 doz. ; lemons, 8 doz. ; pears, 16 doz. 
41. Vest, 65; trousers, 6 10; coat, 615. 48. Bonnet, 65; dress, 613; 
eloak, 617. 43. First, 625; second, 640; third, 635. 44. 61. 
45. B, 40 acres; A, 60 acres ; C, 60 acres. 46. 9 hours. 

Page 44.-17. -15(a + 6). 18. 13 zy(a - 6). 19. S2Vab. 

80. - llmnVx'. 

Page 45.— 87. 5(a + c). 88. 0. 29. Vx. 30. baVxy, 

Page 46.-38. -Aa'^h-\-\bah-\-2ah\ 39. 5ax+2az2-zV- 40. Oz'y. 
41. z« + 3z2y 4- 3zy2 -|- yS. 42. %anx'^ - 3nz8. 43. 8a6 + 46 + 3c. 
44. 7a6 + 56c-3a6c. 45. 3z2y + 6zy". 

Page 47.-8. 16a. 3. 3a + 6 + c. 4. lly-22r. 5. z« + 42«. 

6. -7a2+96y+27. 1. Aay'^+cz^+lO. 8. \1 ahd^+Zb^y^, 9. -QoC^bc 
•^ 22 ab^c - 7 abc^ 10. 2 a^- 2 62+ 7 d^. 11. 24z2+ 12y2. 12. 3pz-n. 

13. _5y2+10l2;2. 14. -3a6+85 6c+2 ac. 15. -14z2+75y2_342.2. 

Page 48. — 16. 23 z2+ 25pxy - 5 y2. 17. 5 aa^ - 4 bz^ + 2 cz. 18. 7. 
19. wn(z2- z + y); 3a(z«-z^-zy3-2j^). 80. z(z - a - 64-c)+ a6 ; 
a%b - c) - 62(d - c). 81. z2(z - 3 y + 5j^2) ; 5(x2 - 2 y) + 6(z + 2 y^). 

22. (a-c)x-\-(b-d)y; a(Vx-Vy-{-V2z). 

Page 50. — 10. l2xy-Sz^-{-Sy. 11. -4zy. 18. 26. 13. 4a«-4rt6. 

14. 2a + 2 6-2c. 15. 3a2-10a6-3c. 16. 5z2+2zy-10. 17. 6z^y 
+2y». 18. -6z2y-2ys. 19. -2a2-362+10. 80. 2z8-2az2-f Sa^z. 
21. 26 + 2d-2e. 88. -2c. 28. -2pv, 24. a24.2a6+362. 85. 8/)% 
- 5pz2. 86. 9x8 + 31 ax^ 4- ISa*. 27. z* - 4z8y 4- 12zV + 2y*. 

Page 51.— 28. 3(z24-y2) + 10(a2_62). 2d. 5y/c^^T^-^Vifi-^-\-5y/S. 
30. z8+z2-y8-y2. 31. -Sxh/. 1. 9(z4-y). 8. ZVx^. 3. (3a+6) 

(z2- y2). 4. 16z2- (a - z^). 5. z - 5Vxy + 3y. 6. 2a2-f 3az + Sz*. 

7. 5z2c-2xc. 8. (a-b)(x-{-y-\-z). 9. -a(5-z). 10. z2+4zy-y». 
11. 3az2-26x2. 12. 7x8y+x2y2+5xy8+7 y* + 8. 13. 22(a + 6)-(a-6). 

Page52. — 14. -2(x2-y2). 15. 22 (0^+ 62) + (a^- 6*). 16.22(a+6+c). 

17. 2c. 18. 6xya + 2y«. 1. a - 6. 2. -a- 2c. 3. 4a2+ac. 



A>JSWERS. 377 

Page 53.-4. -h^. 6. 4a;H-5. 6. -x^-^xy. 7. -x^+xy, 8. 17-a;2. 

9. a2+ 62_ 2 a6. lO. a:*- y*. 11. 4 ax8 + 6a;2 4. feo; + 9. 12. 4 a2a;2 ^. ^xK 

13. 2 a262_ 2 ab. 14. a. 15. a - 6. 16. y. 17. 6 m - 8 n - p. 18. 12 m 

- 8 ri - r. 19. - a - 7 6 + 2 c. 20. 6x - 11. 21. a + 4 y. 22.-6. 

23. a -b-\-d, 24. -a6+6c+c. 26. x. 26. -2+3 a-a*. 27. a+6+7c. 
28. 18. 

Page 56.-2. Sa^c-Sabc-Sac^. 8. 6x^y+6xy^-bxyz. 4. 3rt*6 

- a862 + 3a2&8. 6. 12 x*y2 _ g a;8y8 ^. 20 a;2y4 + 4 icyS. 6. -3ac2x2- 

2 c8x8 4- 6 c^x*. 7. 4 a*6x - 24 a^x^ _ 4 a^f^^* + 4 a^x&. 8. 6 a^y^ - 
6a*x&y2_i5a&a;8y2+5<j2ca;22^. 9. Ga^b*x^-42ab^x^y^-6ab^3^. 10. 6x2/* 
-2a:2y8 + xV- 11- -^a^bd-iabcd-i-iacd^. 12. - | a'-6* - y^^ a^fts 
+ fa868. 18. 2 a.22^ _ J x8y5 + J a;*2^ - I a;y7. 

Page 58. —8. 4 a2 4. 8 «» + 2 a*6 + 3 a* - 5 ««& - 10 a^b^ - 17 a«62 ^ 
150^6'*. 4. 4x6 - 9x* + 31 ajs _ 4a;2 + 27 jc + 16. 6. x^ + x2y _ 6x2/2. 

6. 2 x8 - x2 - 16x + 16. 7. x2 + 7xy + 122/2. g. 6m* - 13 w2» - m2 - 
6n2 + 3n. 9. cfi - a^b\ 10. a^ - 3a25 + Saft^ _ 58. xi. ax2-8a2a; 
+ 16 a*. 12. X* - 2/*. 18. 64x* - 9 a^y\ 14. x* + x22/ - xy^ - 2^. 

16. -m*w2 + 2m8n8 + 3m2n-m2n*-3mw2. 16. x* - 9x8+22 x^-24x. 

17. X*- 2x22/2 +2/4. 18. 2x5-10x*2/+9x32/2-17x2/*+42/^ 19. 25x6a4 

- 30 x^a^ + 14 x4a« - 8 x^a^ + 3 x2a8. 20. a^ + a* - 3 a86 + 3 a^b'^ - a^b^ 

- a*b - tf 68 + 06*. 21. x^-x^y-2 x^y'^ + 2 x^y^ + X2/* - 2/^. 22. x8 + 
3x2/ + 2/8_i. 23. a«-3a6-29a*+ 101a^ + 102a2_644a + 480. 

24. z^-yz*-2y'^z^-\-2y^z^-j-y*z-y^. 26. x6-4xS+3x*+2x8+4x2-16. 

Page 59.-1. x8+l. 2. 1-x*. 3. x3+ 3x22/ +3x2/2 +2/8. 4. a^-Sa^b 
+ 3a62 _ 68. 6. x* - 2x22/2 + y*. 6. a* + 4a2&2 ^ 1654. 7. i5a;4 4. 

16x82/ - 17 x22/2 + 8x2/8-2/*. 8. 1 - a2 4. ^3 _ 2 a* + «&. 9. 26x6 - x* 

- 2 x8 - 8 x2. 10. X* - 2/*. 11. a8 _ j)6, 12. 3 a* - 26 aS^ + 37 ^252 _ 
14a68 + 3o2 _ 6 aft 4. 2 62. 18. €fi-a^-^a*b-a*b^+a^b*-ab*-\-b^-b\ 

14. x5 + 6x*a + 10x8^2 4. I0x2a8 4. s^a* + a^ 16. x^ - x^ + x2 - 1. 
16. 36x*-39x8+13x2-26x+4. 17. x*+9x2+81. 18. x5-41x-120. 
19. 20x - 56x2 + 10x8 - 32x* + x^. 20. 3x8 + Sx^y* + Sy^ 21. a^ - 

3 abc + 68 + c8. 22. a* - 2 rt252 4. 4 ^5^2 + 6* - c*. 

Page 60. —23. ofi-y^. 24. x*-a*. 25. 6*-562c2+4c*. 26. a2-62; 
X2-2/2. 27. 4a2_52; 4x2-92/2. 28. w2_^2. 4w2-n2. 29. x8 - 
3x2-4x+12. 80. x*-2x2+3x-2. 81. x*+2ax8-9a2a;2_2a8a;4.8a4. 
32. a*+2a86_2a68-6*. 33. x*-a2x2-2ax2+2a8. 34. a2-2a6 + 62. 
35. x8 - 3 x22/ + 3 X2/2 - y^. 36. 8 x8 + 12 cx2 + 6 c2x + c8. 37. a* - 4 a^b 
+ 6a262 - 4a68 + 6*. 38. a* - 4 a86 + 6a262 _ 4^68 + 6*. 39. x2 - 2xy 
-2xz + 2/2 + 2yz + ^2. 40. 27 x8 + b\x^y + 36x^2 + 8 2/8. 41. 16x* - 
96 X82/ + 216 x22/2 - 216 xy^ + 81 2/*. 42. 3 d^ + a6 + 6 ac. 43. a8 _ 2 0525 
-4a62-68. 44. 2a6 + rtc. 46. 3a6. 46. a2 - 4a6 -3 62. 

Page 61. —47. 6 a2 4. 2. 48. x* + 3x22/2 + 2/*. 49. 4 X82/ - 4 X2/8. 
60. a8+4a2+ 6a + 2. 61. 0. 
Page 63.-6. -2 6*c. 6. 9a2y2. 7. 6m*wx2. 8. -2ac. 9. 8g. 

10. 40a8x22:. 11.7 m?/. 12. -4a*6. 13. 12x. 14. 8a2a;8y. 16. -7 nr^. 

Page 64.-2. -x^-^xy- 2/2. 3. - a6 + 2 a62 - 3 62 + 6. 4. xy- 
2x22/2 -32J. 5. -x8 + 2«x22/-2a2x2/2 + 2a82/8. 6. 2 2/ - 4 X82/2 - x*. 

7. -6x8+2 ax2- 10 a2x. 8. - 6 x + 4 2/ - 9 z. 9. 6 0^62- 4 a68+ 3 a6c2. 
10. -6x+ 14x2/2^ + 6x82/ -20 2/2. 



378 ANSWERS. 

Page 66. —6. 2 x«+ 4 x^+ 8a; + 16. 6. a^+ Sxhf + 33cy«+ s^. 7. a:H 
ary + yK 8. x*4- y*. 9. 2a^-Sab +46-5. 10. 6a26 - bab^. 11. x + y. 
ia.x-y. 13.3a + 26. 14.6a;-7y. 16.4a;+12. 16. a^ + a6 -f 6-«. 
17. 1+ a: + x2. 18. \i x* + ^ x^y '\- 4 y^. 19. 25 m*- 35 itiH 49. 20. m^- 
niH + mn^ - n'. 21. a* + a^6 + a'^b^ + aft* + M. 22. x* + x^ ^. y4, 
28. 25a:H10a;+l. 24. 36x4- 24aa;24- 4a2. 25. x^- 6x4- 4. 26. x+a. 
27. 9x*- 16x2-1-25. 28. a<- a-^b^-\- b*. 29. 81x*4- 27x^4- 9x2+ 3x + 1. 
80. 144m<+84w2+49. 81.6x2-3x4-9. 82.2x2-6. 38. x24-5x-l. 

Page 67.-84. 1 - y2. 35. a* 4- a;*y2+ «V+ V^- 86. x*- x2y24- y*, 

87. a« - a* 4- 1. 88. 1 - 3 x 4- 2 x* - x*. 89. a'^x* 4- ab^'^y 4- 6*y2. 

40. a 4- ft. 41.4x2-12x4-9. 42. 1 4- 2y 4- 3y2 4- 2y3 4- y*. 48.3x2- 
xy-2y2. 44. 4 4-3x4-2x2. 46. x* - 6x8-11x2 4-99x4- 10. 

Page 68.— 1. j aft2 4- i ft-^c 4- tV 2. 2a2- |a6 4- Jft2. 8. ix--Vy 
4-5-xy. 4. ia24-T^a&4-62. 6. \oi^-^j\x - j\. 6. Ja^- | 024- ^i^a 4- i- 
7. J a 4- J ft. 8. 4x2 - ix 4- T>5. 9. ix8 - -.^^x^ - A^^ + tV 10. ^ a*- 
y\ aft2_ J 63. 11. xs^. 1 a;24- J X- J. 12. i x - |. 18. i a 4- i ft. 14. J x2 
- j^x- J. 16. Jx4- iy. 

Page 69.-5. a* - 2 a2ft2 4- ft*. 6. 2 -5x4- 10x2 - 6x8 4- 5x* 4- 2xS. 

7. 6 x6 - 26 x* 4- 49x8- 66x2 4- 3.3 X- 10. S, x^+x^y*+y\ 9. a^-Qa^ 
4- 13 a* -14 0584. 10a2_4a4- 1. 10. a6-41o- 120. 11. Sa^-6a^ 
-5a4-16a8_ i2a2-6a-3. 12. xS4- i«* -f i^a^ -f Ji*^ - ^x - t\. 

Page 70. — 18. am3^—anx^-\-(bm—ar)x^-\-{cm—bn)3^+(dni — cn—br)x^ 
-(dn-^cr)x-dr. 14. x6-6x5-3x44-40x84- 24x2-24x4-4. 15. a^-Ox^ 
4- 16x*- 20x84- 15x2-6x4-1. 19. x2-3x-l. 20. 2x2 4-3x4-2. 

Page 73. — 4. x2-l. 5. x2-3x-l. 6. X24- x 4- 2. 7. 1 4- x 4- 2 x2. 

8. x8-x2-3. 9. x24-5x-l. 10. x84-3x2y4-3xy24-y8. 11. x2-Jx-J* 
Page 77. — 8. x = 10. 4. x = 5. 6. x = 6. 6. x = 3. 7. x = 6. 

8. x = 4. 9. x = 3. 10. x = 6J. 11. x = 7. 12. x = 6. 18. x = 4. 
14. X = 9. 

Page 78. — 15. x = 1. 16. x = 10. 17. x = 7. 18. x = 9. 19. x = 4. 

20. X = }. 21. X = 6. 22. X = 6. 28. x = 1. 24. x = 6. 25. x = 3. 

26. X = 2. 27. X = 9. 28. x = 14. 29. x = 15. 80. x = 2. 81. x = 6. 
82. X = ft. 83. X = 2. 34. X = 1. 85. x = — 1. 86. x = 4 a. 37. x = 3. 

88. x = 6. 89. x = -. 40. x = 2-a2-ft. 41. x = a 4- ft. 42. x = a4-ft. 

a2 

43. X = 2 ft. 44. X = aft. 

Page 81. — 15. 38 and 65. 16. 6. 17. 22 and 32. 18. Man's a<;e, 
25 yrs. ; brother's. 30 yrs. 19. 15 yrs. 20. Son's age, 9 yrs. ; father's, 
36 yrs. 21. B, $252 ; A, $324 ; C, $424. 

Page 82.-22. Horse, $120 ; buggy, $80. 28. Less, 27 ; greater, 46. 
24. Boys, 78 ; girls, 65. 25. 15. 26. First, 4 ; second, 12 ; third, 24. 

27. Unsuccessful candidate, 1030 ; successful, 1630. 28. A's share, 
$1243; B's, $904. 29. 112 sheep. 30. A, $36; B, $40. 81. 60 pieces, 
32. Horses, 4 ; cows, 12 ; sheep, 80. 33. Half dollars, 60 ; dimes, 40. 

Page 83.-34. 60 children. 35. A, $3500; B, $4000; C, $7600. 
86. Eldest, 24 yrs. ; second, 21 yrs. ; third, 18 yrs. ; fourth, 15 yrs. ; young- 
est, 12 yrs. 37. Less, 7 ; greater, 11. 38. C's share, $440 ; B's, $1320; 
A's, $2640. 39. Women, 18 ; men, 22 ; children, 50. 40. 6 vessels. 

41. First, $ 1300 ; second, $ 1500 ; third, $ 1.300 ; fourth, $ 900. 42. 103 gals. 
43. Sister's age, 6 yrs. ; brother's, 12 yrs- 



ANSWERS. 379 

Page 84. — 44. Artillery, 260 men ; cavalry, 450 men ; infantry, 3800 
men. 46. 10 hours. 46. 50 and 20. 47. $2400. 48. Left pocket, 15 ; 
right, 20. 49. Youngest, $ 425 ; third, $ 475 ; second, $ 625 ; oldest, $ 575. 
60. Quarters, 5 ; dimes, 16 ; five-cent pieces, 75. 61. 45 miles. 62. 5 
miles. 

Page 85.-4. a^ + Oac-f 9. 6. 25 + 10a; -|- x^, 6. 9a^ -\-6ab -f b\ 
7. lQx^-\-Sxy + y'^. 8. 4 a-^2 ^ 12 ax -f 9. 9. 9 a^ + 30 a6 4- 25 &2. 

Page 86.— 18. 25-10a;4-x2. 14. a'^-2a^l^-\-b^. 16. 9a^-Qab-\-bK 

16. a;2 - 10 xy + 25 y^. 17. a^^ - 8 ax + 16. 18. 16 a'^ _ 24 a6 + 9 62. 

Page 87.-19. a^ + 6ax + 9x^. 20* x^-6xy-h9y^, 21. 4m^ + 
4w+l. 22. 1-4 771+4 w2. 28. a^-^a'^b-h^b^. 24. 4a2-12a&+962. 
26. 9 a*- 12a-^62+4 6*. 26. a2x2 - 4 axy + 4 y2. 27. a^fea -f- 4 adc^ -f- 4 c*. 
28. a«62 - 4 a^bcx + 4 c2a;2. 29. a^y^ + 50 a:*y + 625. 80. 16 a2ft2 _ iq ^bxy 
+ 4 a;2y2. 83. 9 x2 _ 9. 84. 4 a2 - 9 62. 35. aH^ - 62. 36. 1-9 x*y2. 
87. 16 a*x2 - 9 yK 88. 25 a*6* - 1. 

Page 88. —41. x"^ -^ y'^ ■{■ z^ -\- 2xy - 2xz - 2 yz. 42. x'^ -^ y'^ + z^ - 
2xy + 2xz-2yz, AZ. x^ -\- y^ -\- I -h 2xy - 2x-2y, 44. x2 4. yS _|_ ;j2 
-2xy -2xz-\-2yz. 46. 4 a2 + 62 + c2 - 4 a6 + 4 ac - 2 6c. 46. a2 -f 
962 + c2 + 6 a6 - 2 ac - 66c. 47. x2 + ^2 ^ ^3,2 ^ ^2 4. 2^2/ + 2 x;? + 2rx 
+ 2yz -{-2vy -\-2vz, 48. x2 -f- y^ + ^2 4. ^2 _ 2xy -2xz -2vx +2yz-\- 
2vy-\-2vz, 49. a2+4 62+c2+4cZ2_4a6-2ac+4ad+4 6c-86<?-4c(i. 
60. a2 4- 62 + c2 4- ^-^ + 2 a6 - 2 ac - 2 a(i - 2 6c - 2 6(i 4- 2 cd. 61. a* + 
6* 4- c* 4- <«* 4- 2 rt262 - 2 aV.2 - 2 a^d^ - 2 62c2 - 2 62(f2 + 2 c2d2. 62. a* 4- 
64 4. 4c2 4. d2 4- 2a262 - 4a2c + 2a2d- 4 62c 4-2 62^-4 cd. 

Page 89. — 68. x2 + 7x 4- 12. 64. x2 - lOx 4- 24. 66. x2 4- 6x - 24. 
66. x-«-2x-63. 67. x2-3x-18. 68. x^-8x-65. 69. x2-16x4-60. 
60. x2 - 3x - 108. 61. x2 - 24 X 4- 135. 62. x2 -(a + 6)x + a6. 

68. a2 4- 5 a - QQ. 64. 81 - 9(x + y)-{- xy, 66. x2 - (2 a + 3)x + 6 a, 
66. x2 + 2 ex - 15 c2. 67. x2 - 4(6 4- l)x 4- 16 6. 68. x2 + a6x - 2 a262. 

Page 90. —69. 3x2 - 26x + 35. 70. 6x2 - 11 x - 35. 71. 12 m^ - 
31m4 20. 72.3x2-10x4 3. 78. 3x2-6x-2. 74. 5x2 -47 x + 84. 
76. 6x2- xy- 22/2. 76. 6 - 13 a6 + 6 a262. 77. 2 a2 - a6 -362. 

78. 5x2 + 6 xj/ - 8 y2. 79. 2 w2 4- mH - 3 m2n2. 80. 3 a2 + a6 - 2 62. 
81. 3x2-66x-262. 82. 10 2/2 _ 12 a?/ 4- 2 a2. 88. 4x2 4- 4x2/ - 3v2. 
84. 15 a2 4- 4 a6 - 35 62. 1. x - j/. 2. a2 4- 6. 8. x - 3. 4. 3a4-*6. 
6. 2ax+3. 6. bx-y. 7. 3a-2 6. 8. 2x2-32/2. 9. x+8. 10. a-10. 
11. X — 2a. 12. x-3c. 

Page 91. — 18. x + y, 14. x2 - 2/2. 16. x2 + 4. 16. 3 x - 3. 

17. 2rt+36. 18. 3ax4-6. 19. 10+3a6. 20. l-\-Sx% 21. 5ax2-l. 
22. 4 x^y 4- 3. 

Page 92. —28. x^ - xy + y^. 24. x^ -{• xy + y\ 26. 4 a2 - 2 a6 + 62. 

26. 4 a2 -h 2 a6 4- 62. 27. x^ + 3 6x 4- 9 62. 28. 1 4- 2 2/ + 4 2/2. 29.4 2/2 4- 
2 2/ + 1. 80. x22/2 + xyz + z\ 81. x^y^ - xyz 4- z^^ 82. 9 - 3 xy 4- x'^y^. 
83. X* 4- «22/2 + y4. 84. X* - x^y^ 4-2/*. 88. 1 + 4 x?/ 4- 16 x^y^. 86. 9 x;^ 
+ 6x2/ +4. 



380 ANSWERS. 

+ 2x2-f-a". 47. x*-2a5*+4 058-8x2+ 16 a;-32. 48. 7?+Sx^-h9x+27. 
49. 4x*^ - 2x 4- 1. 60. x^ - Sxy + 9y2. 1. x'^ + 2xy-^ y^. 2. 4x2 _ 
4xy + y^. 8. f»^4-2m+l. 4. l-2w + m2. 6. 4-12 a H-Qa*. 
6. 4x2-12xj/ + 9y«. 7. 9x*-12x2y2 + 42^. 8. a* + 4a26+4 63. 

9. 25 - 10 he + 6V. 10. 4 rt2x2 + 4 axy2 + j^*. n. a%'^ - 2 a6c + c^. 
12. 49 - 28ax + 4a2x2. 13. c-^ + (P + c^ + 2cd + 2c« + 2dc. 14. a^ + 6' 
+ c--2a6+2ac-26c. 15. x^-^y^-^ z^ +2 xy -2 xz -2 yz. 16. l+c2+(P 
+ 2c-2rf-2cd. 17. 9a2+ fc-* + 1 -6a6 + 6a-26. 18. a2 + 46« 
+ 1 -4afe + 2a-46. 19. x-* + 4y^ + 9 -4xy + 6x - 12y. 20. a* + 
4 52 _(_ 9c-s H- 4a6 + 6 ac + 12 6c. 21. 26 a^+^Hc^-lO a6-10 ac+2 6c. 
22. r7i'''+nHr^+s^+2mtt-f2»nr+2ms+2nr4-2ns+2r5. 28. m^+n^+r* 
+ «2 _ 2 wn + 2 mr - 2 w« - 2 nr + 2 n« - 2 rs. 24. a^ + b^ + c^ + I + 
2 a6 - 2 ac - 2 a - 2 6c - 2 6 + 2 c. 

Page 94.-26. x2 - y^. 26. ??i* - n^, 27. 42/* - 1. 28. 1 - 9aV. 
29. 26-30x + 9x2. 80. 49 a^ + 70 x + 25. 81. 16a* -462. 82. x* - y*. 
88. x6-y8. 84. a-H3a_28. 86. x2-7ax+12a2. 86. x2+(6-c)x-6c. 
87. x2 — (m — 71 )x — mn. 88. 9n2 + 7 w — 60. 89. wi2 -f (r + »)w + rs. 
40. x3+l. 41. x»-l. 42. x*-l. 48. a»+x». 44. 3fi-a\ 48. x^+B. 
46. x8-27. 47. x*-y*. 48. x»-j/*. 49. x^-xy-\-y^. 60. a2^.aa;-rx*. 
61. x8 + x2y + xy^ + y8. 62. «« _ ^2^. + aa;-2 _ jgS. 88. a2 4. 4 „ 4. 16. 

64. a8-3a* + 9a-27. 66. x2 - 3. 86. 1 - a2. 87. «« + a2 + a + 1. 
68. x*+x8y+x22/2+ 3:^84.^4. 59. 4x2-2xy+j/2. go. 4a2+6a6+962. 

61. 3 ax - 4 62. 62. 2 mH - 6. 68. a + 3. 64. x - 4. 66. a2 _ 52. 
66. x2 - y*. 

Page 96. —11. 5a8a;2^ 5a8x2. 12. 9x2y*, 9x2y*. 18. UmH*r*, 

11 m2n<r8. 14. 7 wim2s6, 7 mn^s^. 18. 8 a68c2, 8 a68c2. 16. 12 xy2r2^ 12 xyz^. 

Page 97. — 8. 5a6(2a+6). 4. 7 xyz(2x—9y), 8. ax(x2— axy+y2)^ 

6. a^h(a'^-ab-{-b^). 7. o2(flfa;2 4. y _ i). g. 3a26(3a; - 6y - 1). 
9. 6x8(3y-2?/2 + 0). 10. 4x*y(l - 3y - 4y2 4. 2 y«). 11. 3x2y2 
(X - 2 x2 4-3 2/2-4 xy). 12. ay'^^l - 3 a^ + 6 a^z^ - aV^)- 

Page 98.— 2. (2x+y)(2x+2/) 8. (6 + x)(rt+y). 4. (a-y)(a+x). 

8. (2/-6)(a + x). 6. (x-2m)(y-n). 7. (a + 6)(x2-y2). g. (y'^+1) 
(2/ + I). 9. (a-6)(a24.3). 10. a (2/- 6) (x2- 2/2). 11. (3a+2)(2a2-3). 

12. (2a2-6)(3x2-2 2/2). 2. (2 x+2/)(2x+v). 8. (a+3 6)(a+3 6). 
4. (a2_6-2)(a'i_52). 5. (3a;_2/)(3x-2/). 6. (2x2-5 2/2)(2x2-5 j/2). 

7. (x-l)(x-l). 

Page99.— 8. (2?/-l)(2?/-l). 9. (3x-4 v)(3x-4 2/). 10. (5 x- 2/2) 
(5x-v2). 11. (l2x2-6 2/)(12x2-5?/). 12. (1 - 5 a22/2)(l - 6 a22/2). 

13. (5 + 3a6)(6 + 3a6). 14. (2-10 rt6)(2-10 a6). 18. (xy^-S)(xy^-S). 
16. (4 + 5a6-^c3)(4 + 5a62c8). 17. (11 a + 100 6*)(ll a + 1006*). 
18. (m2n-20«2)(wi2w-20s2). 19. (rt4-6-2)(a + 6-2). 20. (x-y-S) 

(X-2/-3). 21. (m2-6n)(m2-6?0- 28. (a + 6)(a - 6)(x 4- 2^) 
(x-\-y). 24. (x + y-{-z)(x-^y-^ z). 26. {x - y -\- z)(^x - y -^ z). 
26. (a2_26 + 2)(a2_26 + 2). 

" Page 100.— 4. (30254.4 c) (8 a26- 4 c). 6. (x2/+2 2/«)(x2/-2 2/-?). 
6. (x24-2/2)(x + 2/)(x-v). 7. (3 v+l)(3 2/- 1). 8. (2ax+6a)(2(ix-()a\ 

9. (4rt2x24-7 2)(4a2.y2_7;j). 10. (1 4-l)0-2)(l 4-3^)(l -3;?). 11. (y'^-^iz'^) 
(y-\-2z){y-2z). 12. {U-^a'x*)(\\-a^x'). 13. (x2+7 2/22f)(x2-7 ^2^). 

14. (12 + 5rt2/)(12-5rt?/). 18. (x-^^a~h)(x'^- a-\-b). 16. (a+X-y) 
(a-x + 2/)- 17. (2x2 + a + 6)(2x2-a-6). 



ANSWERS. 381 

Page 101. — 18. 4xy. 19. (a + 6+c+(?)(a+6-c-d). 20. — 4mn. 
21. (a-6+c-(?)(a-6-c-frf)- 22. Sab{a^-^b^). 23. (5a-l)(7-a). 
24. (5a-6)(a + 56). 26. (Ox - y)(4a; + 6 2/). 29. (2 a - b -\- xy) 
(2a-b-xy), 30. (2 x-^ + Ssc - 1)(2 a;'^ - 3x+ 1). 81. {Qab-c) 
l-4ab-c). 32. (a - 6 + 2 a:y)(a - 6 - 2 xy). 88. (1 - x + 4 aft^) 
(l-x-4a62). 34. (« - 2 6 4- 2x + 3y)(a - 2 6 - 2x - 3y). 36. 
(a+l46-c)(a+l-6 + c). 36. (x-y4-w4-»)(ic-y— w— 7i). 37. 

{x -\- y -\- s — z){x-\- y — 8 •}■ z), 38. (a — c + 6 + d)(a — c — 6 — d). 

Pagel02. — 41. ix'^-\-y^-^xy)(x^+y^.-xy). 42. (x24-3?/2+2xy) 

(x2 + 3 y2 _ 2 xy). 43. (m^ -2n^ + 2 mn) (m^-2n^-2 mil). 44. 
(a* + ft2 4. a'^b){a* 4- 6^ - a^6). 45. (x*+3 y* + 2 x2y2)(a:4_f.3 y4_2 a;22/2^. 
46. (2x2-3 2/2 4.2xy)(2x2-3y2_2xy). 

Page 103. —48. (x2 + 2 + 2 x) (x2 + 2 - 2 x). 49. (8 + y2 + 4 y) 
(8 + 2/2-4y). 60. (x + 4)x. 61. (x2 + 2 y2)x2. 62. (a2 + 2 62 + aft) 
(a^^2b^-ab). 63. {a^ - b^ + 2ab)ia'^ - b'^ -2ab). 

Page 104.-3. (x+5)(x4-2). 4. (x-ll)(x-2). 5. (x-9)(x-5). 
6. (x + 4)(x + 6). 7. (x+7)(xH-8). 8. (x-7)(x-4). 11. (a; + 8)(x-3). 

12. (x-12)(x + 5). 13. (x-9)(x + 5). 14. (x+9)Cx-7). 16. (x + 14) 
(x-4). 16. (x-10)(x+7). 17. (x-9)(x+8). 18. (x+7)(x-6). 

Page 105. — 19. (x + 4) (x + 4). 20. (x + 10) (x + 4). 21. (x + 9) 
(x+7). 22. (x+12)(x-f6). 23. (y+12)(y+8). 24. (x-9 a)(x-6a). 
26. (x-5a)(x-3a). 26. (x-7)(x-8). 27. (x - 1.3)(x -4). 28. 
(y-20)(y-7). 29. (x+7)(x-4). 30. (x4-9)(x-2). 31. (x+11)' 
(x-6). 32. (x4-12)(x- 11). 33. (x + 14)(x - 13). 34. (xy - 11) 
(xy+2). 36. (xy-|-8)(xy-13). 36. (x-12 a)(x+5 a). 37. (x-11) 
(x+9). 38. (2-17)(^+16). 39. (x4-25)(x+7). 40. (x + 16)(x-6). 
41. (x-20)(x-20). 42. (x + 13)(x - 12). 43. (y4-ll)(y + 6). 44. 
(x-19)(x-4). 46. (x+3)(x-18). 46. (y+14)(y-3). 47. (y-17) 
(y+lO). 48. (y4-18)(y + 6). 49. (x2 + ll)(x2 -f 6). 60. (x - 2) 
(x2 + 2x + 4)(x8'-7). 61. (x4+ ll)(x2 + 3)(x2-3). 62. (y + 2) 
(y2-2y+4)(y-l)(y2+y4-l). 63. (yS-16)(y6-3). 64. (x+6)(x+c). 

66. (x + a)(x-c). 66. (y -h a)(y - a){y + b)(y - b). 

Page 106. — 68. (x-16)(x+16). 69. (x4-16)(x+15). 60. (y+23) 
(y-13). 61. (y-30)(y+ 16). 62. (x - 27)(x - 20). 63. (x - 21) 
(x+20). 64. (x+36)(x-35). 66. (x+30)(x+8). 66. (x+40)(x-33). 

67. (x-f-24)(x-23). 

Page 107.-4. (3x-7)(x-6). 6. (3x+6)(2x-7). 6. (4m-5) 
(3m-4). 7. (3x-l)(x-3). 8. (3x+l)(x-2). 9. (5x-12)(x-7). 
10. (3x+6)(x-2 6). 11. (3x-2y)(2x4-y). 12. (6x-4y)(x + 2 y). 

13. (2r7i + 3mw)(m-wn). 14. (3 a-2&)(a+6). 16. 2(y-o)(6y-a). 
16. (2x-y)(2x4-3y). 17. (5x + 7 y)(3x - 5y). 

Page 108.-20. (x+3)(3x+5). 21. (x4-2)(7 x+6). 22. (5x-12) 
Cx-1). 23. (2x+3)(3x-2). 24. (x-2)(3x+6). 26. (3x-8)(2x+6). 
26. (2x-3)(3x-2). 27. (x -f 3)(3x - 4). 28. (x-4)(3x-6). 
29. (x + 3)(10x-7). 

Page 109.-2. (x + 5) (x + 4). 

Page 110.— 6. (x+3)(3x-4). 6. (x+3)(x+4). 7. (2x-7)(x+4). 
8. (3x-7)Cx-2). 9. (2x + TO)(x-3»i). 10. (3x + n)(x- 2n). 



382 ANSWERS. 



112.— 6. (a+2)(a;2_2a; + 4). 6. (x-l)(x^-{-x-{-l). 7. (!+«) 
(1 - X + ar'-*). 8. (X - '6){x^ 4 -^x -f 9). 9. (5 + x-)(25 - 5x2 ^ x*). 
10. (4 + x-^)(2 + x)(2 - X). 11. (x2 + 16) (X + 4)(x - 4). 12. (a^ - 3) 
(«*+ 3aH 9). 13. (x-2_ y)(x4+ x=^y + y'^)- 14. (m 4- n)(w2- mn + n*) 
(?/» - 7i)(m2 -f- m/i 4- n^). 16. {m^ + »'^)(w< - mH^ 4- »*♦). 16. (x - 1) 
(x*4-x8 4-x2 + x+ 1). 17. (1 4a46)(l -a-6 + a24.2a64-62). 

18. ra - 6 4 l)(a2 _ 2a6 + 6'^ - a 4- 6 4- 1). 19. Smn(m^ + n^). 

20. 2 n(3 m2 + n^. 21. 3(a 4- x)(a - x)(5 «« + 8 ax 4- 5x^). 

22. (a-2x)(7aM-8ax4-4x2). 23. (x + y + v^xy)(x + y- V2xy) . 

24. (a 4- b'^ + V2 «6-^) (a 4- ft^ _ V2a62). 25. (?»2 + n^ + V 2 m^n* ) 

(m2 + n* - V2 wj2,i4)^ 26. (9 4- x2 + Vl8 x2) (9 4- x* - VlSx^) . 

27. (a*4-6*4V2a*/>4)(a»4-6*-\/2tf464). 28. (3 x2 + 2 y + Vl2x^) 
(3x2 4- 2y-Vl2x22/). 29. (2 Xj4 y^ 4- >/4x2/2)(2 x 4- 2^^ _ V4xy2). 

30. (x2 4- 1« + V32X2) (x2 + 16 - V32'x2) . 

Page 115. — 7. (x-l)(x4 2)(x4-3). 8. (x 4- !)(« + 2)(x -3). 

9. (x4-2)(x4-3)(x-5). 10. (x - l)(x - 6)(x 4- 7). 11. (x - 2) 
(x-3)(x- 5). 12. (x-l)(x4-3)(x + 7). 13. (x - 2)(x - 2)(x + 3). 
14. (X4- l)(x-3)(3x-4). 16. (x - l)(x - 2)(2x - 3). 16. (x4- 1) 
(6x2-6x4 13). 17. (x4-3)(2x2-6x+ 13). 18. (x-l)(x-2) 
(x4-3)(x-4). 19. (x4-2)(x-2)(x2 4x4-4). 20. (x + 2)(x-2) 
(x2- 3x4-1). 

Page 116. — 1. (3x-l)(3x-l). 2.xy{x-\- y)(x + y). 8.2x(x-2) 
(x-2). 4. 5a2(x-3yO(«-32/'0- 6. x(x-6)(x-5). 6. 12 ox(ax -4 6) 
(ax — 4&). 

Page 117. —7. 3a(m-w)(w-n). 8. 5a(a— 6)(a-6). 9. (x'-4-l) 
(X4-1). 10. 3a&(x2-2/2_|-8). 11. 7x2(x-3v)(x-32/). 12. a(a4-l) 
(a-1). 13. {2xy-{-z)(2xy-z). 14. 7x2(x-3y+2xy^)(x-3y-2xy2). 
16. bahy\x 4- ay)(x 4- ay). 16. 9(a2 4- 2)(a2 - 2). 17. (x2 4- l)(x + 1) 
(x-1). 18. {a^^b'^){a^h){a-h). 19. {x^+y^){x^-\-y^){x^y){x-y). 
20. (6+3x8)(5-3x8). 21. (l4-w»2)(l4-m)(l-m). 22. {a-h-\-c-d) 
{a-h-c-\-d). 23. 4m». 24. x(2x 4- 3)(2x - 3). 26. 5s(x2 + 3y2) 
(x2-3y2). 26. 5(x-v)(x+2/). 27. (,n^-^h^-\-^a%^){n^-^b^-9a'^b'^). 

28. (5x-8)(x-2). 29. 3x22/2(x+32/)(x-3 2/). 80. 3ay(x-3y)(x-3y). 

31. 5x(a- 6 4-3x)(a-6-3x). 32. (1 -3x + 2xy2)(l - 3x - 2xi^2), 
33. (x4-d)(x4c). 34. (x+d)(x-c). 35. (x— n)(x— wi). 36. {a-d) 
(6 4-c). 37. (2c-d)(a-2 6). 38. (3x - y)(« - 32:). 39. (x + 3) 
Ix-y). 40. (l4-a-6c2)(l-a4-&c2). 41. (a2+ 024- ac) (024- c2-ac). 
42. (x2 + 1 + x) (x2 4- 1 - X). 43. (wi2 4. 7i2 4- mn) (m^ 4- n2 - mn). 
44. 4(x2 4-2 4-2x)(x2 4-2-2x). 45. 9(a2 + 2 4- 2a)(a2 4- 2 - 2a). 
46. (w2 4-8n2 4-4win)(m2 4.8n2-4mn). 47. (x2 4- 2 y2 + 2 xy) 
(x24-2y2_2xy). 48. (2m2+n2+2m7i)(2m2+n2-2win). 49. (2a4-36) 
(2a -36). 60. (x2 4- 2^/2 + X2,)(x2 + 22/2 _ ^y). 61. (x2 4- V^ 4. xv) 
(x2 4 2/2_x2/). 62. (a2 4-2 62)(a2_2 62). 53. (2 4- a2)(2 - a^). 
54. (l-x)(l4-a;4x2). 56. (2-2/)(44-2y4-y2). 56. (x--y)(x24-xy4-y2). 
67. (a4-6)(a^-a6 4-62). 58. (1 + a)(l - a 4- a^). 59. (x-1) 
(x2 + X 4- 1). 60. (m 4- n)(w - n)(?7i2 - mn 4 n^){m^ 4- wn + n*). 
61. (2a2-362)(4a*46a262 + 96*). 62. (3+x)(9-3x+x2). 63. (a24-2) 
(a* - 2 a2 4- 4). 64. (a2 + 62) (^4 _ ^-ifta + ^4). 65. (a 4- 6)(a - 6) 
(a2- ah + 62) (aS + a6 + 62). 66. (a - x) (a* 4- a^x + a2x2 4- ax« 4- x*). 



ANSWERS. 383 

Page 118.— 67. {x-\-2)(x-2)(ai^-2x-\-4)(x^+2x-^4). 68. (l + m) 
(l-m)(l-m+w2)(l+w + 7»2). ^9. (;3_a:)(y+3x+a;-i). 70. (l + x+y) 

(1 -x-y-^3(^^+2xy-\-y^). 71. (x - y + l)ix^- 2xy -\- y^- x -\- y -^ I), 
72. (x««-ll)(x+l)(x-l). 78. (x'^-7)(x-fl)(x-l). 74. (a+3)(a-3) 
(a4.1)(a_l). 76. (x+12)(x-10). 76. (x+24)(x-3). 77. (x+16) 
(x-15). 78. (x-18)(x+8). 79. (x+17)(x+13). 80. (7x+l)(3x-l). 
81. (4x- 7)(9x+8). 82. (x - 2l)(x - 20). 83. (x - 15)(x - 12). 
84. x(x-5)(x + 4). 85. (4a-2)(a-3). 86. (4a2 _ 2)(a2 - 2). 
87. (3x+13)(x-5). 88. r>(x^-2x-b). 89. (x-7)(x+l). 90. (x-5) 
(x-5). 91. (x-7)(x + 2). 92. (x2-12)(x2+6). 93. (X'^-lU)(X'^+4). 
94. (3x+2y)(3x+2 2/)(3x-22/)(3x-22/). 96. (x+n)(x-n)(x2+w2). 
96. (x+a+l)(x+a+l) 97. {a-{-y){a-y)(a^-ay-\-y^). 98. (x2-3m) 
(xH2). 99. (x + y-2?)(x + 2/-2). 100. (a- 6 + c-fd)(a- 6 -c-d). 
101. {m-n-^a-b)(m—7i-a-^b). 102. (a-2 + w-n)(a— 2 — wi+w). 

103. (3x-2/+2s+22)(3x-y-2»-2i?). 104. Cx-y+l)(x-y4-l). 

106. (ax - &)(x 4- y)(x + y). 106. (x2+ y^) (^^ ^. y) (a; - y) (a - 6)(o -6). 

107. (m-n)(x-y)(x-y). 108. (x-l)(x-3)(x-5). 109. (x-l)(x+l) 
(x + 3)(x-5). 110. (x + 6)(x + 2)(x + l)(x-l). 

Page 121. — 6. 25. 7. 15 ab^c. 8. 8 xV- »• ^ a*«V. 10. x-y. 
11. x-1. 12. xHax2. 18. 15(x-l). 14. 4(a2-62). 16. a^-ab-\-b^, 
16. x-1. 17. x2 + x+l. 18. x2-4x + 4. 19. y + 2«. 20. x2 + y. 
21. X + 4. 22. X - 9. 28. X + 7. 24. x - 12. 26. x + 1. 26. x2 + 12. 

Page 122. — 27. x-1. 28. x-5. 29. 2 x + 3. 80. x - 1. 

Page 126.-6. x-5. 7. 2 x + 3. 8. x 4- 5. 9. x - 4. 10. x - 3. 
11. X- 6. 12. X- 4. 18. x2-9x + 21. 14. x2 + x+l. 16. x + 3. 
16. X2-X-5. 17. x2-4x+3. 18. 2x-5. 19. 2x2-x4-3. 20. x^-y^. 

Page 128.— 4. Iba^y. 6. I2exh/^z^. 6. im a^b^<^c^x'^. 7. (x+y) 
(x-y)(x2-fx2/+y2). 8. 35xy2(a+5)(a4 6)(a2-a6+62). 9. a'^(a-b) 

(a-4 6)(aHa6+62). 10. (x-2)(x-5)(x+5). 11. (x+10)(x-ll) 
(x-9)(2x-5). 12. (x+5)(x+3)(x-l). 13. (x-12)(x+7)(x+5). 
14. (7x-l)(x-5)(x+5). 16. a&(rt+6)(a-6). 16. (x2-f y2)(x+y) 
(x+2/)(x-y)(x-y). 17. (x-l)(x+l)(x2-x+l)(x2+x+l). 

Page 129.-3. H.C.F., 2x-3; L.C.M., (3x-4)(12x2-4x-21). 
4. H.C.F., x-5; L.C.M., x(x-5)(x2+5x+2)(7 x2+x+l). 6. H.C.F., 
x+1; L.C.M., (a;+l)(3x2-llx-l)(6x2-17x-7). 6. H.C.F.,x-l; 
L.C.M., (x-l)(3x2 + 3x-2)(21x2 + x-10). 7. H.C.F., 4x-3; 
L.C.M., (4x-3)(x2+4)(3x2-2x+4). 8. H.C.F., x+1; L.C.M., 
x2(x+l)(x2-x-l)(x2+x+l). 9. H.C.F., x2-6x+6; L.C.M., (x-2) 
(x-3)(x-l)(x-4). 10. H.C.F., 2x-3; L.C.M., (2x-3)(3x-2) 
(x + 4)(3x + 4). 

Page 132.-1. -|^. 2. ~ 8. -?^. 4. -^. 5. x + y. 

36 a 3 m x + xy 3m2 

a q8 4- a ^b + ab^ 4- 6« 

D. • 

a-b 

Page 133.-6. i^. 6. ■^. 7. -l^m^. 8. 2i±2^ 

5 6 4 6x lip 3 6 

3 4axy ^ jQ 2(a-6), ^^ x+_2 ^^ _«±12_. ^3 a;4-9 ^ 

3x2 -8y 9(a4-6) x-9 x(x-7) 4a(x-4) 



384 ANSWERS. 

14. ^:^^^ 15, -JLtl 16. gV12x+144 j^ x±a^ 

X- - 9 X + 20 x2 + xi/ + y2 x + 11 ' x + c 

18. ?--«. 19. ^ ^'tf . ' aO- ^^- 21. ^-. 22. ^'-^+^^ 
x-c x*+x*V+y* a-^+x-^ x-a a(a+x) 

28. -^^±1-^ 24. ?^±^. 
4(x + 6) 3x-6 

Page 134.-26. ^]-T^-^^ , ae. 2£^. 27. _£JL5 

88 «^ + 3a:2 gg x--^4-3x+27 3^ x^-6x+25^. 31 x2+3x-7 
x(4a+3x)" ' x2+3x+3' ' 2x2-3 x+ 15* * x2+3x+16* 

82. ^'-^^ . 88. ?!^l2x4:_5. 84. ^^. 

x'-^ 4- « + 1 x2 + X - 2 X 4- y 

Page 135. — 8. ax - b. 4. 6 + ?. 6. 6-—. 6. x^-y^. 

a a 



7. X + -^ — 8. x2 4- xy + 2^2. ^, a + b + -=-^. 10. x - 2 - 

x-{- y a — b X— 1 

11. a-x + -^. 12. x+2y-^l^^. 18. x2-5ax+-^. 14. 3+— ^• 
a+x x-\-y 3x x2— 1 

16. a-b. 16. x2h-x2/ + 2/^ ^. 

x-y 

Page 136.-6. ^^^M:^. 6. ^^^'-^°. 7. ^i^. 8. 2o^Jzl«. 

4/ t/ X "4" t/ X 1 

a 2 ax ,rt 2a8-aft-fa2ft_262 a;3 x3-3xJ2^ + 3 0:22 

S>. - • lU. ; • 11. • Is. • 

a+x a-\-b x+y x — z 

IS ^y 14 2^Llli!5?. 

' x + y ' a4-x 

Page 138. -6. -«i, A 4- 6. A^, -|i-, -|^. 7. -^. 
^ rtfec a6c a6c 12 ac 12 ac 12 ac 10x2 

25 4x g 4q8-4a2& 3a2H-3aft 2 62 ^ qxy-f5xy ay-\-^y 

10x2' 10x2* * 12 a8 ' 12a8 ' 12 a^" " x2y2 » 3.2^2 ' 

«^_+-?.? 10 ^+ ^ 1 X- 1 ji q2 _ 2 a& + 62 a + 6 

x2t/2 * • X2 - 1' x2 - 1' X2 - 1* . ' d^ - b^ ' CC^ - b'^' 

q8 + q62 4- q26 + 68 ^g ax 4- 3a q2x4-2q2 2q 

a2-62 ' ■x2 4"6x4-6' x2h-6x + 6' x2 4-5x4-6* 

,0 x8+3x2-3x-9 3x + 3x2-x3-9 x2 -- a2x24-a2M2 

id. . • — — • 14. : '^—l 

x2-9 x2-9 x2-9 x*-y* 

62x2 - 62y2 a-b 



y\ j#4 /g4 yA. 

Page 140.-8. 1^. 4.^^^ + ^°- 6.^=^- 6. «'-2»y+y'- 

5x 2x2 x^ xy 

- 4a2x4-6q2_-a;a ^ q6x 4- x2 ^ 2 x2y« - 6 y* -t- 6 y^ -I- a^' 

4q2x2 * ' a6y * * 10x2ya 

10. y'-^y-^. 

xy 



ANSWERS. g85 

Page 141.-12. -^. 13. -2^,. 14. ^i^+i^. 15. -1«^. 
^ x^-y'^ a^-b'^ x2_y2 cfi-b^ 

jg xy-2y-3x + 9 ^^ 5a;^-13x + 8 ^g 8a ^^ 3(x-l) 

x2 - 6a; + 6 " ' x^ + a - 12 * * 1 - 4 a^' ' aj2 _ 4 * 

20. -^(7^+^). 21. --t«^. 22. -J-. 23. «^(«-±^. 24. -^. 
jc-2_2a;-15 a^-b'^oc^ a-b a^ - b^ x-a 

26. 4+^!- 26. 0. 27. ^{^\^^^-^') , 28. -i^. 29. " ^ 



aa-62 a2-x2 a;^.y x^+ax+a^ 

30. A«i.. 81. y'-^y . 82. ^'-<^\ 83. «H«^6-axHto^ , 

a^+fts x^+arV+y* ax{x-y) ax(a^-b'^) 

84. a^ + «^>-e5a + 606 3^ 2^y^. 3^ ^^^ 3^ 2(x + 1) ^ 

rt2 - 62 X* - y4 a- 

33 2a34.q62_^2q2ft_ft8 3Q Q ^ m8+mnHn8 ^^ 36 

a2ft_ft8 • • • • (w+»)« ' * (x+3)2(x-3)' 

Page 142.-42. ^^i±i^^. 43. ?-+?«. 44. ^ 



a;2 + 2x-15 a -3a l-x 

45 a^ + 9ax ^3 8 ^- x (x^-Sx + S) 

(x + a)2(a-x)*  (x2-16)(x-3)' " (x2_ I6)(x-6)(x- 3)' 

48. ^^-^^-^^ 49. '-^ 



x3-10xM-31x-30 (a; + a)(x+ 6)(x + c) 

Page 143.-51. -«^±^. 52, ^-i^±J^^ 53. «^-4a6-62 

ax(a-6) a2-62 a4_2a262 4.ft4 

64. i 65. — 56. 1. 57. 0. 

(a — c)(c — b) (x — a){a — b) 

68 x2y + x y2 + 2xgg-2y2g g^ 3 x^ - a^ - 6^ - c^ 

(x + 2/) (x - ^j) (y + 2) * (x — a) (x - 6) (x - c) 

Page 144.-3. — 4. -• 6. a - &. 6. — — 4. 5^^. 
^ 6 x2 a- 6 10^2 

6. • 6. — • 7. a(x — 

5 2 ^ 



Page 145.-8. -L-. 9. — «^,. 10. ^ia±iL^. 

a + x (a + 6)2 x2-xy + y2 



11. ?. 



12. '^±y. 18. ^^ti. 

X — 2/ X — y 

Page 146.-4. ^. 5. -«^. 6.-^^2^. 7. -« • 8. «' 



Say 2x2y2 2(x-y) 3x a- 6 

9. x2 - y2. 10. ? 11. xy. 12. x + 1. 13. £fizL^. 1. ^, 

^ 3(6 -- c) a + 6 3 

«>!».>. « 21x ^ 3rtv R 35ac2x2y2 ^ 662 - a-x ^ a 

2. 4a6c. 3. -— — 4. — •-• 5. — mr^-' 6. -• 7. • 8. -. 

10a 2x ib^s a-b a x 

g 3Ca8 + a62 + a26 + &8) ^^ 2(x - y)2 
6a ' 6y 

whitb's alg. — 25 



886 ANSWERS. 

Page 147.- U. ?i«^«^±^. 12. —T-^ 7- 18.-- 1*- L 

15. « - ^ 16. iBLlJ^. 17. _i 18. ^-^^^^ 19. — !_ 

aO. (x+w)«. 21. ^ 22. ^±^. 28. ^^. 24. 1. 

25. I ., ae. (^' - y'^)^ 27. — ^. 28. -«^. 29. i^+5!. 

80. x'^ + y^. 81. (a%2_i)2. 

Page 149.-4. H 6. ^. 6. .^ ^  7. ^±^. 8. — 51±i— . 

2/ x^+a;+l «— a ox+x— a 

9. ''-+"■ 10. a- 1. 11. ''(«-*). 12.x. 18. -L-. 14. "-+6. 
a a(J) — x) 1 + a b 

18. «»-«J'. ie.«i^!!zdi. „.k:i«^. 18. a-l. 19.?^±ii3?. 
ax 6(<i'^-l) l-3o y" 

" a« - 6'^ o - 1 a* + 1 a+b 

«1+_L_. 8. "' + '''' . 7. 3a + 2 6. 8. ^+^. 9. i^. 



a*H-a"^+l tt^ + ax + x" x + 3 2x + 3 

10, -^?L. 11. ^J_. 12.1. 18. -«-. 14.-^. 16. 2 



a-6 a-x X a + 1 x'^— a^ a(a2— 1) 

16. ^>«-2 . 17. -ini_, 18. i_. 19. ^y+^'y' + ^^-y*. 

ai/ x2 -^ 4 j/a x2 - 1 x* - y* 

20 ^^>a^'^ - f 30ax-24aa g^ 8 q^x - 4 ax' + 4 gS - a^ H- x^ gg x^^ 
20 a(a + x) ' * ^a^x^ ' ' y^' 

28. ^^^^^^ 24. 5f^. 26. «. 26. a(a2 + ft^). 27. ^^. 

28. «'(^ + V). 
a — h 

Page 151.-29. ^i^^t^. 80. <^fta;-xy + x^ gj xy« + a;^^ + y^z^ 

10 a6y xy^ 

88. _?y_. 88. li(^±ii. 84. ,^-.'» 85. . „^"' „ . 

x'-* — 2/*'* X wr — m^ — m m* 4- 3 win + 2 n' 

3g mHn?, 87.-^^. 88.^^-^:1^. 39. 2^(«zilVi«(«±l). 40. ^^^±^ 
2mn aH6^ ay+6a; 2/(a+l)-x(a-l) 2w»a 

41. 1. 42. -2^. 48. LzJMjx?. 44 2^^ 

a* -f ic* 1 + a y^ + 2 xj/ — x* 

Page 153. — 6. x=60. 6. x=3yV 7. x=6. 8. x=36. 9. x=12. 

10. X = 0. 11. X = 7. 12. X = 9. 18. X = 12. 14. x = 9. 16. x = 5. 

16. X = 10. 17. X = 4. 18. X = 6. 19. x = 8. 20. X = 1. 21. x = 1. 
22. X = 1. 28. X = XO. 



ANSWERS. 387 

Page 154. — 84. x = 2|. 25. x = 2f. 26. x = 3|. 27. x = IJ. 

. x = --4i. 29. x = -2. 30. x = 0. 31. x = -2. 32. x = 6. 

33. X = IJ. 34. 2/ = 55. 35. x = 2. 36. x = 4. 37. x = 8. 38. x = 7. 

39. x = l. 40. x = 2. 41. x = 4. 42. x = l|i. 43. x = 4. 44. x = 0. 
45. X = f . 46. X = 2. 

Page 155. - 3. x = -^. 4. x = t-^^— 5. Hrn + d ^ c) 

a — b 3a-f26 a 

e. x=«iftz:^. 7. x=2!=i*!. 8. :,= «»'(»+") . 9. :c- ^°'-'''^''+''> - 
a— c 26 w— a 4a6 

10. x = - ^^-/>'' 11. x = a-6. 

w + n 4- i> + »• 

Page 156. — 12. x = a + b. 13. x = 14. x = ^ ~ ^^ ~ ^' . 

* 6a + 26 a-6 

15. x = • 16. x= — : — 17. x= • 18. x = 



6-d b 26 — a a + 6-2c 

19.x = «!^t^^±^. 20.x = ^?LziJ?. 21.x = i?L. 22.x=i>gr. 
a+6 w+n 2n 

1. 72 and 60. 2. -^^^ and -J^^^^ 3. 24, 36, and 48. 

n 4- w « 4- wj 

Page 157. — 5. -, ^, and -• 6. First day, 27 mUes ; second, 36 

6 3 2 

miles; third, 24 miles. 7. Eldest, $3000; second, $ 2400 ; youngest, 

$ 1800. 8. First, ^ 280 ; second, 1 2 10 ; third, $ 246. 9. Carriage, $ 88 ; 

horse, $187. 10. Horse, $120; saddle, $20; bridle, $10. 11. B's age, 

50 yrs. ; A's, 40 yrs. 12. Husband, 30 yrs. ; wife, 18 yrs. 13. 3y^j days. 

Pagel58. — 15. 4 J days. 16. 1? days. 17. 6Jf days. (2) A, U^^V 
days ; B, 18 A days ; C, 34^ days. 18. 24 days. 19. 24 days. 20. $56. 
21. First, $3258; second, $2896. 

Page 159. —22. Smaller, $3240 ; larger, $4860. 23. First, 60 yards ; 
second,' 46 yards; third, 40 yards. 24. 11 ^ hours. 26. 20 days. 
26. 60 sheep. 27. 36 persons, 60^ ; 18 persons, 76^ ; 46 persons, $ 1.60. 
28. $ 1000. 29. Longest side, 36 rods ; second, 27 rods ; third, 18 rods. 
30. 40 eggs. 

Page 160. — 31. 10 and 11. 32. Men, $12; women, $9; children, 
$4.50. 33. B, $1860; A, $2160; C$2000. 34. C, ^^^; A, ^; B, 

^ ~ ^ ^ 35. 640 yards. 37. $ 46 and $ 76. 38. 76 and 60. 39. ^^ 



and 



3 ., w . ^_^ 

bm 



a— b 



Page 161. — 40. 60 apples. 41. Pears, 40; lemons, 28; oranges, 8. 
42. 8 children. 43. Barley, 18 bushels ; corn, 24 bushels ; oats, 30 
bushels. 44. 1059 men. 45. Five-cent stamps, 9 ; two-cent, 16 ; one- 
cent, 26. 46. 30 acres, 47. 21 days. 



388 



ANSWERS. 



Page 162.— 48. ?L±_5?! days. 49. 4§} hours. 60. C, 40 days; A, 

a + 6 
120 days ; B, 60 days. 61. 50 yrs. 68. (1) lO^f minutes past 2 ; (2) 21^^ 
minutes past 4. 68. (1) 49^j minutes past 3 ; (2) 10}^ minutes past 8. 
64. 1^1000. 66. 36 miles. 66. 6^ ^ouis, 67. 14^ miles. 

Page 163. — 8. 4 p.m. 

Page 164.— 4. 7.30a.m. 

Page 165. — 8. First, 360 miles; second, 190 miles. 10. 12, 18, 5, 
and 45. « 



Page 166.— 11. 12, 28, and 60. 

mn 



18. 



m 



2 + n 



-fi, 



m 



2 + n 



+ n, and 



2-f n 
Page 167. — 16. 187 pounds. 17. 42%. 18. 4000 pounds. 
Page 168.— 19. $28.94. 80. 8%. 81. Syr. 11 + mo. 88. $620. 

83. 3yr. 9 mo. 84. $280.74+. 

Page 172. — 6. x = 13 ; y = 5. 6. a = 8 ; y = 6. 7. x = 7 ; y = 5 
8. X = 10 ; y = 6. 9. x = 24 ; y = 4. 10. x = 11 ; y = 5. 11. x = 8 
y = 5. 12. X = 8 ; y = 6. 13. x = 12 ; y = 18. 14. x = 50 ; y = 20 
16. X = 16 ; y = 12. 16. x = 3 ; y = 2. 

Page 174. — 19. x = 5 ; y = 4. 80. x = 3 ; y = 7. 81. x = 7 ; y = 1 
88. X = 6 ; y = 2. 88. y = 14 ; « = 10. 84. x = 11 ; y = 7. 86. x = 4 
y = 2. 86. X = 6 ; 2? = 12. 87. x = 10 ; y = 9. 88. x = 6 ; y = 4. 

Page 175.-81. x = 12; y = 8. 38. x = 2 ; a? = 3. 88. y = 4 ; « = 5 

84. X = 3 ; « = 2. 86. y = 5 ; 2: = 1. 86. x = J ; y = i^. 87. y = 14 ; 
= 10. 88. X = 11 ; 2? = 13. 89. X = 16 ; y = 12. 40. x = 7 ; « = 3. 

41. X = 12 ; y = 6. 48. x = — ; y = ?• 48. x = 16 ; y = 12. 

a b 

Page 176.-1. x = 13; y = Q. 8. x = 2; y=3. 8. x = 11 ; y = 2. 

4. X = 8 ; y = 4. 6. x = 4 ; y = 3. 6. x = 10 ; y = 5. 

Page 177. — 7. x = 5 ; y = 6. 8. x = 8 ; y = 10. 9. x = 12 ; y = 9 
10. x = -l;y=2. 11. x = 6;y=12. 18. x=-f;y = V. 18. x = 3 
y=2. 14. x=J; y=l 16. x=7; y=3. 16. x=10; y=2. 17. x=38} 
y = 70. 18. x = 6;y=12. 19. x = 11^ ; y = 4J. 80. x = 3;y = 5 
81. x=9; y=123}. 88. x=d\', y=2j\, 88. x=5; y=8. 84. x=6 
y = 4. 86. X = 13.1 ; y = 5.6. 86. x = 5 ; y = 3. 87. x = 4; y = 5 

88. x=8; y=2. 89. x=^^i^^^; y^^kiiln^, 80. x=tt^ ; 

^ 2 

Page 178.-81. x=^-^±-^ y=«^^. 88. ^(f"^^') ; ^(-"F^- 
* a+6 * a+6 m^ - n^ -'* -^ 



w 



-n2 



33 ^_ mn(an-\-bm) , mn(am-bn) ^ ^_ ll(a+6) . 9(a-6) 
m^-hn^ to2 + n2 * ' 2 ' ^ 2 ' 



86. x = 






86. x = 



ab^ 



1 ' ' wj2»2_i a2+6^' 

J^^-^. 87. X = «i^^l26) K2a^L^. jg. ^^ = iiL» • « = !!. 



ANSWERS. 389 

Page 179. —40. x = 4; y = 6. 41. a = 4 ; y = 6. 42. x = 3 ; y = 5. 

43. X = f ^ ~ ^% y=J*!.Zl^. 44. a;==3jy = 2. 45. « = 7 ; y = 6. 

o» — am an — bm 

46. as = -^^ ^^ ; y = -^^ — .-^^« 47. « = ; y = 



nq — mp * np — mq ' a + b* • a — b 

Page 181. —6. x=S] y=2j e=l. 6. a;=7; y=5; 2j=4. 7. a;=3; 
y = 4; z = 6, 8. x = 2; y = 3; 2r = 4. 9. x = 2; 2^ = 3; 2: = 7. 
10. X = 2 ; y = 3 ; 2r = 6. 

Page 182. —11. x = 5; y = 4; 2: = 3. 12. a; = 8 ; y = 4; z = 2, 

13. x=ll; ?/ = 9; z = 4. 14. a; = 6; y = 12 ; 2? = 20. 15. x = 24 ; 
y = 60; 2 = 120. 16. a; = 3 ; 2/ = 4 ; 2? = 6. 17. 3C=7J; y=7 ; ;?=-lJ. 

18. a;=y=g= ^ ^^^ » 19. a;=-4— ; y= ^ ; «=t-^ 

aft + ac+oc a+o— c a— 6+c b—a-\-c 

20. sc = 4 ; 2/ = 9 ; 2; = 16 ; tJ = 25. 1. 147 and 196. 2. 65 and 24. 
8. Boy's age, 18 yrs. ; girl's, 7 yrs. 

Page 183.-4. A's age, "^ + ^^^ ; B'sage, 9±^. 5. First, $2500; 

m—n m—n 

second, $7500. 6. Silver, 562 ounces ; copper, 60 ounces. 7. Nickel, 

20 ounces; copper, 28 ounces; silver, 12 ounces. 8. 24. 9. A, $300; 

B, $ 600. 10. Tea, 86 cents ; sugar, 8 cents. 11. Tea, ^^* ~ ^^ ; sugar, 

9IL^^£m, 12. f. 

ad — be 

Page 184. — 18. if 14. y. 15. 1 and 6|. 16. Upper, 45 inches ; 
lower, 63 inches. 17. Finer, $1.20; coarser, $.80. 18. 10 and 2. 

19. A's age, 49 yrs. ; B's age, 21 yrs. 20. Persons, 13 ; sum, $ 3. 

Page 185. — 21. Larger, 10^ hours; smaller, 14 hours. 22. 9, 15, 
and 24. 23. 276. 24. A, $3|^ ; B, $3 ; C, $2^. 25. First, $8 ; second, 
$ 18 ; third, $ 16. 26. Wheat, $ 1.26 ; rye, $ .95 ; oats, $ .80. 27. 2, 5, 7, 
and 10. 28. First horse, $ 56 ; second, $ 33. 

Page 187. — 3. 2^12. 4. y4«. 5. a;"'. 3. 2iSa^^ofiy^^. 4.^. 5. — . 

Page 188.— 6. 64a^b\ 7. ^^. 8. 27 a^b^ 9. 25a*xV. 

27 a^ 
10. - 64 o866a^. 11. - aio66c6a;io. 12. 729 aisfti^co. 13. - 8 a^^. 

14. 81 a*b^^y^. 15. 0656^4. le. a^b^c\ 17. 2«a«'»ar"V"% when n is 
even ; — 2'»a'*"a;«V% when n is odd. 18. ^'^''a^b'^x^, 19. x^y^z^, 

20. - a2»+i62«+ic2n+i. 21. - 27 y^'if^. 22. ^^^. 23. - A^^£-. 

^ 16 2^2 125 «»&« 

24. ??^^. 25. «:^^: 26. «?^^. 27. -^^. 28. - ^— • 

Page 190.-1. oi^-bx^yJt 10 a^j^ - 10 x22/8 + 5 ^.^^ _ 2^. 

Page 191. — 2. a« - 6 a^ft + 15 a*62 _ 20 a%^ + 15 a%^ - 6 afe^ 4. 56. 
8. TO* - 4 TO»n + 6 TO2n2 - 4 wm^ + n*. 4. a^ + 5 rt*x + 10 aH^ + 10 a^x^ 
+6ax*+x5. 5. a5-5a*x+10a8x2-10a2x34-5ax*-x6. 6. x"'+7x62^ 
+ 21x62^ + 35x*2^ + 35xV + 21x22/« + 7x2^ + 2/^ 7. c* - 4 c^d -f 6 c^d* 
-4c(P + d*. 8. a^ + 6a66-f 16a*62 + 20a868+ 15a26* + 6a6fi+6«. 



8d0 ANSWERS. 

9. l-3a; + 3a«-x». 10. x* - 4x» + 6a5« -4x+ 1. 11. -56ii«6». 

12. 120 c^cP, 18. -Mofiy*. 14. a«- 8 0^6+28 a*^* . 15. c»+9c»<l 

+ 3a c^rf^ + .... 16. 2^0 - 10 a»y + 45 zV . 17. •• + 16 m^ii* 

-Qmn^-^n^. 18. .. -f 28a%»+8aa;7^a*. 19. 21 a^&^+Tafc^-ft'. 

Page 192. — 21. 27 o» - 64 a^ft^ + 36 aM - 8 6». 22. (fi-9a*bc 

+27 a262c2-27 6»c«. 28. a*M-2a«68c+§a26-2c2_ja&c»4-— • 94. — -ia^ftc 

16 16 

+ i a'^b^c^ - 2 ab^c^ + Mc«. 25. 32 m^ - 80 m*n* + 80 iii«fi« - 40 iii*ji» 

+ 10 mn^ -n^. 26. ^- 3 cC^b^c + 24 abc^ - 64 c». 

Page 193. — 8. 25a*«V. 4. 216 a^b^e'^x^, 5. -32aWxiS!f» 

6. 81 mi%Vy*. 7. -p^zV^. 8. d^xiV^^ 9. — - 10. ^^ 



tns 81 nV 

11. — ^^ ^ . 12. — - — ^ „ ' 18. ^=-^ — , when nis even: — . 

when n is odd. 14. ^'*,^'' . 15. m* + 4 w»«n + 6 w^n^ 4. 4 „i«8 ^ „4. 

16. m*-6TO*n+10m8n2-10w2n8+6TOn*-n6. 17. a»-3a*6+3a262-6», 

18. z*-4a;«4-6 3c2-4z+l. 19. l-Saj+Sx^-x*. 20. l-4z^-^6g* 
-\7fi-\r^' 21. a:»-9x6a2+27«3a4-27a«. 22. yi2_9y8;j2^.27y42:4-272«. 

28. 16a* + l6a8a; + 6a%2 + ax8 + i^. 34. «?^ _ «!^ + aft^i _ c». 

16 27 3 

25. a;*-4x2a+6a2-i^+^. 26. a«4-^+^4--- 27. xiH16xi*y 

x^ X* a^ a* a» 

+ 105xi8y2.|. ... +106x2yi8+16xyiHy^*. 28. x^o - 10 x^a^ + 45 x^o* 

+ 45 x2ai« - 10 xa>^ 4- a'^. 29. x^^ + 12 xS^y + 66 x^V 4. ... ^. ee x2y«> 

+ 12 xy" + yi2. 80. - 792 aV. 81. - 252 x^. 82. - 126 x*. 

Page 194. — 1. 5 ax^. 2. 2 x^y^r*. 8. 4 m^nx*. 

Page 195.— 4. ±5a6. 5. ±9rt2&8c4. e. ±4x2y. 7. 4a62c». 
8. -3 77in2. 9. ±2a62a;*. 10. -6mn2»*. 11. ±Zxhj^z, 12. -3a2c*. 

13. -2o2a;. 14. ab, 15. a«x2. 16. ±x«y». 17. x*y8. 18. ±^-^- 

6xy2 

19. -««^. SN>. -2^*. 21. ±1«^. 82. «7?. 88. *3f. 

6 62c* x2y 4w8n* y^z «• 

24. - 



X*"y2n 



a26"» 
Page 196. — l.x-2y. 2. 2x2+y. 8. x2-3y2. 4. 3 a -2 2/2. 5. 2a-l. 

6. l + 3y8. 7. 3x-|. 8. ^-y2. 9. x + y-2. 10. x2+ 2y2 ^. s^g 4. 4. 

Page 197. — 12. x2- 2 xy + y2. ig. 3 a;2- 2 xy + 5 y*. 14. a2- 5 a - 6. 
15. 1 - 6 X + x2. 16. 2x - xy + J y. 

Page 198. — 17. 1 -2x + 3y. 18. 9 - a + 6. 19. -ix2+ |y + |. 
20. Jx-li/. 21. x-y + |. 22.H-x4-y + «. 28. 1 -x + x2- x8+ x*. 

24. a»-3a + ?-^. 



ANSWERS. g9l 

Page 202. — 14. 263. 15. 307. 16. 240. 17. 459. 18. 702. 19. 723. 
SO. 626.. 21.13.3. 22.6.72. 23. .094. 24. .025. 26.8.68+. 26.24.22+. 

27. .238. 28. |. 29. I6h SO. 32}-. 81. .591 + . 32. .73+. 83. ||. 
84. j\. 86. 1 .414 + . 36. l". 732 + . 87. 2.236 + . 88. 1 .870 + . 89. 2.569 + . 

40. .948+. 41. .774+. 42. .816+. 48. .935+. 44. .845+. 46. .516+. 
46. .741 + . 

Page 203.— 8. x2 + 2a. 4. 2«-a. 6. 5a8-362. e. a^ + a + l. 
7. x2 4.23c -4. 

Page 204. —8. 2x'^-x-6. 9. l+3«-3a;2. 10. x--- 11. a-l+-. 

X CL 

12. 3a2 - 4a6 - 262. 13. 2x2 _ ^xy + y\ 14. 3x2 - 2ax + a\- 

Page 206. — 8. 551368. 9. 314432. 10. 456533. 11. 175616. 

12. 1124864. 18. 1953125. 

Page 207. — 16. 35. 17. 57. 18. 145. 19. 364. 20. 325. 21. 301. 
22. 4.2. 23. 4.6. 24. .46. 26. 12.5. 26. 4.07. 27. 3.04. 28. \, 
29. \%. 80. If. 81. 3^. 82. .712 + . 88. 1.908 nearly. 84. 4. 
86. 1.259+. 86. 2.08+. 87. 2.15+. 88. .873+. 89. .941+. 40. .793+. 

41. .427 + . 42. .464 + . 48. .411 + . 

Page 208. — 44. 2 a + 3 6. 46. x2- 2x - 2. 46. x + y. 47. 1 + 2 a. 
Page 211.-1. y/cm. 2. V(x - y)^. 8. yl^- 4. Vcfiofi, 

6. y /{x + y)2. 6. \{a-x)J. 7^ \a"6«. 8. Va62. 9. Vx^. 

laJIH 11.^. 12.4/1^11. u,V^, 14. A^. 

^x + y 'X* ^(x — y)8 ' ah 

Page 212.-16. \^I6626, v''256, \/iOOO. 16. v^, v^,_v^. 

^ n* 'n'» ^9 ^a ^h ^c 

20. v^, v^xiy2, .jgy. 21. v^ (x-y)8, v ^(x+y)2. 22. V(a-6)2, V^T^T, 

Va=^. 28. v^oiofts, ^^, ^ ^^\\ 1. 6\/3. 2. 4v^. 8. 4\/5. 

^a* ^(g + 6)^ - 

4. 3\/3. 6. 4a62v^. 6. \baby/bb^K 

Page 213. — 7. 5aV6 a~2 6. 8. 2v^2xgy 2-33^ . 9. (a+6)Va^. 
10. (x-y)Vx2 + xy + y2. n. (x + y) v^x - y. 12. (x2-y2)\/S. 

13. 26(a2_25)V3. 14. 5a(l-y)v^. 16. (a:2+xy)v^^. 16. JV2T. 

6 2 

17. i\^. 18. J v^. 19. v^. 20. 2aVl5x. 21.2av'l5a6. 22. "Vox. 

28. — ^ — V5(a2-62). 24. _2«_ V3(a2 - 62). 26. -J_Vx2^. 
a2-62 J q- 6 x-y 

26. (x + y)V^r^. 27. v'x^'^H:. 

Page 214.— 28. VIO. 29. Vbax. 80. \/a23p. 31. (a+6)Va2-62. 

82. v'(a+6)(a-6)='. 88. \^2(a-3 6). 84. yVx(x-y). 86. v'a'-«6(a+6). 

1. \/8ax. 2. VOOtf^. 8. -^^. 4. -^-^^-^ 6. v^2ax. 6. \/9x^. 



392 ANSWERS. 

11. va - 6. If. VI - a«. IS. y/-{a-b)b. 

Page 215. — S. 7V2. 4. 17V6. 6. ISVS. 6. -7*5 V30. 7. W^^. 
S. 19v2. 9. 7\/6. 10. 2OV0. 11. -Jv^. 12. }V2. 

Page 216. — IS. |iV3. 14. |Jv^. 16. Vv^. 16. - 3v^l0. 

17. iv4. IS. iva. IS. fl + l:-l^^/3. 90. fV2. 21. ay(3a;-6y) VxT 

a \a h xj I 

99. 0. 9S. 3((i» - ^)vW. 94. 2qHh|0ax^ ^ q 

96. (6a -5ac- 2 aa:)>/a^^ "^ 

Page 217.-4. 168. 6. 50\ /6. 6. f v^. 7. ^v^. 8. iV6. 9. JVlS. 
10. 120 v^4. 11. 12 vl8. 19. \/337600. 18.6^^64. 14. 24v^. 16. v^. 

16. ^yj-^' 17. 12 aft v^. IS. 6xV5ay. 19. ISoxWdx. 90. Ca^cy. 

21. ?^^V30. 99. eal^y/^^. 9S. 12V^^5^- 24. ahcx^Vii 
10 ^ a-x 

96. a62^'864a5S^. 96. 16 aft v^x^ - y-*. 97. -5v^. 

Page 218.-99. x - Vx - 6. M. 6x - 8 V3x - 24. 81. 6 + \/lO. 
89. 2 a - Vab - 3 6. 88. 16 a + syaft + 6. 84. xVx - yVy. 85. a"^ V3 
+ 3a2_a-l. 86. 2^+6\/6 + 2\/4. 87. V2 4- | VO - j v/3 - 3. 
88. 4a -6. 89. x-4y. 40.33. 41. ix-12. 49. Vx* - 21. 
48. —X. 44. 2x. 

Page 219.— 8. 6V2. 4. ^^Ve. 6. jVS. 6. jV3. 7. f. S. f. 
9.2. 10. J\/36. 11. i\/l8. 19. iv/4320. 18. v^40. 14. 2v^5. 15. 6aVS'. 

16. ^</^. 17. -^V^x, 18. A^. 19. ?^Vi^. 90. V^cT^f. 
6x2 2x 4x 2x^ 

Page 220.— 1. fV3. 9. jVl6. 8. 6\^. 4. iV7. 6. 2V3. 6. iV2'. 

7. 3^. g. _A_V3xy. 9. — \/2^. 10. AV6. 11. ^Vy. 19. -Va6. 
3xy 2a ^ y b 

Page 221.-18. 2V7. 14. 3V2. 16. 2. 16. 2V3. 17. J VS. 

18. 3V2-2V3 jg T-VV30. 20. ^^t^. 21. VIO + 3. 22. 3 VlO 

6 3 

-3V3. 28. VIO-Ve: 24. I±^. 26. 6+2V6. 26. q+2V^ + 6 ^ 

2 a-6 

2^ 30+13V6 gg 6V70~2V2T g^ 6 - V6 3^ 2~ V2+V6 

19_ ' * 238 * * 5 * * 4 

81 2 -f V2 -f VO 32 12 + 9V3 + 3 V5 - 6 VT5 

4 ' 22 



ANSWERS. 393 

Page 222.-36. .707 + . 86. 5.7735+. 87. 1.4433+. 88. .9622+. 
89. -6.121 + . 40. .464 + . 41. .169 + . 42. .127 + . 48. 1.5892 + . 

44. 1.6.32 + . 45. 6.854 + . 46. 2. 8. 16. 4^25V3. 6. 2V3. 6. 2v^. 

7. 4v^. 8. x^ V^, 9 . a^W. V^. \^ a^h^yJ ah\ 11. a*3c*. \%,\^aWy^, 

18. 256 (a - 6) \/a - h. 14. x»Vy. 

Page 223. —16. 2. 17. V3. 18. 5V2. 19. 5v^6. 20. yfa. 
21. Vx. 22. ^yJa^hx. 28. a-i\/a^. 24. v^o^x. 

Page 224.-3. \/6 + l. 4. jV38 + iV2. 6. 2 + ^3. 6. iy42 
-jy2. 7. 2V5 + 1._8. V6 + V2. 9. 3-V2._ 10. VJ + Vy^, or 
JV2 + JV3. 11. iV35-iV5. 12. iV3 + jV2. 18. V5a + Va. 

14. — r-^ , or 3( V3 - v^). 16. ^ -, or \\(cM. + \/2). 

V3 + v^ |V34-^V'2 

Page 225. — 17. VS + v^. 18. 1 + \/5. 19. V6 + V6. 20. 3- \/3. 
21. V5+VTI. 22. V3-\/2. 

Page 226. — 4. x=32. 5. x=9. 6. x=5. 7. x=25. 8. x=2. 
9. x=16J. 10. x=6. 11. x=12. 12. x=26. 18. x=4. 14. x=5. 

16. x=4. 16. x=27. 

Page 227. —17. x=6a. 18. x=^^^. 19. x=0. 20. x=2. 

2 h 

21. X = a2 - 2a6 + h\ 22. x = ?^. 28. x = «1±-^. 34. x = ^. 

16 26 4 

25. X = 2. ' 26. X = a. 27. x = Va2~=n. 28. x = «i^^li}!. 

46 

29. X = ^* ~.^^^ 80. x=10. 81. x = 4. 1. (} xy - J ax*) Vxy. 

2. -^Vx2 - y2. 8. (x+2a)\/a. 4. J\/6-j{/^. 6. -v^l08. 
x+ y _ 6 

6 y/ a^-h 7 x^-{-xy-x\/y-yy/y g (6x-ax) Va6- q6(x2- 1) 

x^-2/ ' ' b^ x^-ab 

9.x-^y/¥^l. 10. v^^^-y^-^ . 11 x+21-10v^34 ^^^^-^ 

y x-29 _ ^ 

18. V3 + 2v^. 14^fV2 + i>/J4. 16.- i>/6+iV30. 16. 2V5-1. 

-- 2V3 + 3\/2-\/30 -0 v^ + V2-2 ,^. 2\/6 on V7 - 3 

17.  • 18.  19. — -—• 20. . 

4 2 ^ _? 

Page 228. — 21. 12\/l6200. 22. -2xy-y^. 28. o*-a-2\/a6-6. 

24. _J_Va2-62. 26. x = ± i\/3. 26. x = /f^^  27. x= ^ ^^ 



a-6 ^ 4 62+C4 c2+l 

28. X = a — 1. 29. x = a + b. 80. x = 9. 

Page 229.-1. 2V^n^. 2. 5v^^. 8. 16V^. 4. 4ay/^^. 

5. x*V^n;. 6. ?V-n. 7. — V^. 8. ^\/^. 9. a2x>/^. 
2 4 5 

10. (a+ 6)V-1. 



394 ANSWERS. 

Page 230.— 3. WV^, 4. 2>/3T. 6. t\V^. 6. lOv'^n^. 

7. 7V-1. 8. jV-i. 9. 2xy/^n, 10. -V^^. 

a 

Page 231.— 8. -VlG. 4. - 15\/6. 5. -432. 6. - 512V3. 
7. -J. 8. WVab. 9. - 6\/^^. 10. - 6 V-T. 11. V^^. 
12. iVlU. 18. -V-'6. 14. -f. 15. -V-6. 16. - 1. 

Page 233. — 17. 17. 18. 7. 19. 2a2+6+6>/^. 30. 6(c2-a2). 
21. -V'G-2 + 2V3-f 2V2. 22. f^. 28. x- + J. 24. fts _ ^8. 

25. 6 + 2 V6. 26. L^ii^^^-^^. 27. l^v^. gg. «!:^^. 

89.6 + 2V6. 80. ^ + ^^^ + ^ 3^ 6 + 10V32 ^ 82. i«P^. 

a — 6 o3 a- + x^ 

88. . (Q a;'-^ - 2) V^ . 34^ 2v^^. 85. 2 + 3V^^^. 86. 7-6^^=^. 
x'^ + 1 

87. _7 + 4V3. 88. -30+10\/5. 89. 4a\/^. 40. -z^-2zyVzy-y^, 
41. -1. 

Page 236. — 1. a*6*c. 2. 3a% 8. o-ift-'c. 4. x^f/^z\ 5. 8a*6. 

1 ^ ? xv2 

6. a^x ^2/2. 7. 2 a"6«c. 8. x»2/»2r3. 9. ^bz'^. 10. aftVy*. 11. -^• 

12. -^. 13. ab^c^^y^. 14. 6*xi. 15. -^. 16. a563. 17. ahfyi. 

18. (-27) J. 19. leJaaa;!. 20. at X a?. 21. at ^ at. 22. aJft'ic"!. 
28. asftl + a'^l 24. (a*62^a^6-2)K 25. y/a^, 26. TiTT* ^- 1" 

28. - ,. 29. -J-' 80. — . 81. Va. 32. — -. 38.4. 84.64. 85.9. 

Va y/a " c^Vb 

86. I. 87. j/^. 88. 3. 89. 4. 40. y^V^* ^1- 1^- ^- i- 48. 13|. 
44. 8. 

Page 238.-5. a^ - 6 J. 6. a J - 6"J. 7. x2+ 2 + aJ"^. 8. 2 + x"2j/-i 
+ x^'2/'. 9. x-y. 10. x-i-1. 11. x24-2x^ + x-4. 12. l + x-*+x*. 
18. a^-ftK 14. a^-aibs-\-bh 15. 9x"J-6x"5y"i+4jr''. 16. x-x3j/i 
+ x^yt-y. 17. a4-&. 18. z^-^x^y^+y^. 19. 2a*-6i 20. x'^+y"*. 

21. x^-2x*-x*. 22. x-2-ix-iy-i + y-2. 23. a'^ or a^. 24. S*-- 1. 

25. 2/^. 26. a-26"'2%-i2 or — !^ 27. a«x«+*. 28. — or 6*. 3* 

a^b 2 ci2 ^ 

Page 241.— 4. x=± 9. 5.x=±2. 6. x=±3. 7.x=±4. 8.x=±3. 
9. x=±6. 10. x = ±7. 11. x:^±l. 12. x=±V^^. 18. x = ±7. 

14. X = ± 2 v/5. 15. X = ± v^. 16. z = ± 3^/3T. 17. x = ± 3. 

18. x = ±vTr. 19. x = ±2>/:rT. 20. x = ±V2. 21. x = ±^jjvl6i. 



ANSWERS. 395 

Page 242.-88. x = ± f . 88. x = ± Vm-\-n. 84. x = 0. 85. x = 

^ ,Vn^. 86. x = ±YV7. 87. x = j: ^ >/6^^4cA 88. x = 



l-a2 " 6-2c 

±iV62c*-2a6c. 89.x=±i. 80. x=±5. 81. 12 and 20. 88. 6 and 21. 
c 

83. Width, 26.7+ rods ; length, 356+ rods. 84. Width, 24 rods ; length, 

60 rods. 85. 10 and 6. 86. Father^ s age, 40 yrs. ; son^s, 10 yrs. 

87. -Vab a,nd- Vab. 88. Son's age, -Vabm ; father's, -Vabm, 
ha h a 

Page 243. — 8. 7 and 2. 3. — 1 and 5. 4. 6 and 4. 5. } and — 2. 

6. — J and J. 7. h and — c. 8. a and Vft. 9. — Va and Va. 

Page 244. — 18. x = 4 or — 3. 13. x = 9 or — 5. 14. x = 7 or 3. 
15. X = 8 or 4. 16. X = — 7 or 4. 17. x = — 8 or 7. 18. x = 14 or — 4. 
19. x = - 13 or -2. 80. x= 22 or -7. 81. x = 2 or -6. 88. x=-f 
or 5. 83. X = I or - 6. 

Page 245. — 84. x = — J or |. 85. x = } or 1. 86. x = | or 5. 

87. X = — a or — &. 88. x = a or 2 a. 89. x = a or 30. x = — Va 

2 

or - Vb, 31. - 3. 38. 2 and 3. 33. - 5 and 2. 34. 12 and - 6. 

35. — 1 and - 4. 36. — 1 and 5. 

Page246.— 39. x2 + 4x-5 = 0. 40. x2 + 7x + 12 = 0. 41. 2x» 
+ 3x-2 = 0. 48. 12x2 -a; -1 =0. 43, 6xa + 7x + 2 = 0. 44. »» 
+ 4x = 0. 45. x2-5 = 0. 46. x2-3 = 0. 47. x2 - 4x+ 1 =0. 



48. a;2+(a-|]x-^' = 0. 



Page 248. — 4. x = 5 or 3. 5. x = — 2 or — 10. 6. x = 3 or — 7. 
7. X = 15 or — 3. 8. X = 19 or 1. 9. x = 4 or — 22. 10. x = 4 or 3. 

11. x = 5or-J. 12. x = 4±V^. 13. x = 10 or - 8f . 14. x = -2 
or — 3. 15. X = — 5 or — 42. 16. x = 4 or 3. 17. x = 7. 18. x = 7 or 

- Vt^- 10. X = 15 ± 4\/l4. 80. X = x^f ± A>/67- 21- a; = ± 6. 
82. X = 5 or - 2. 23. x = ± 5. 

Page 249. — 8. x = 4 or — |. 3. x = 2 or J. 4. x = — 1 or — J. 
5. x = 3 or 1. 

Page 250. — 6. x = — J or — f . 7. x = 2 or - |. 8. x = 9 or — ^. 
9. X = 7 or - 5. 10. X = 4 or — y-. 11. x = 3 or J. 12. x = y^^ or — \. 
13. X = Jjf or - f . 

Page 251. — 16. x = - J or f . 16. x = 6 or - JJ. 17. x = 4f or — 8. 
18. X = I or J. 19. X = - I or - ^. 20. x = J or |. 21. x = 4 or — jj. 
88. X = 2 or - \}. 23. x = 4 or - 4|. 24. x = 2 or - J. 

Page 252.-3. x = 37 or - 13. 4. x = 48.2173 or - 7.2173. 5. x = 17 
or 7. 6. X = 4 or — J. 

Page 253. —7. x = 7 or - 26. 8. x = 6 or - J. 9. x = 2 or 

-6i. 10. x= -^or -4 . 11. x = 5 or -J. 18. x = 1 ± jV^. 

13. x = ^iL±:^i^Sj6m. 14. ^ = zl1±^^. 

2m 2m 



396 ANSWERS. 

Page 254. — 4. x = a or 1. 6. a: = - or — 6. x = a* or 6*. 7. x = 2 

1 rt b a 

or--. 8. ;J(liV5). 9. x = iiorl. 10. x = aor ^ — 

a 2 ^ a-fl 

11. X = a or 6. 12. x = ^aft ± Va^- 6-^ + 4 a^fe*. 18. x = or -(a+2). 

14 X = a ± — 16. X = 2 a or — 6. 16. x = a or - (6 + c). 17. x = c 

a 

-\-d±Viicd-2c^. 18. x = -^or ^A_. 19. x = TOori. 

a -f- a + 6 m 

Page 255. — 1. x = 17 or 4. 2. x = 14 or — 13. 8. x = | or — 1. 

4. x = 2or|. 6. x=§. 6. x = 6or-6f 7. x = j| ± ^^^ Vl3. 
8. X = a or 2 a. 9. x = i or — 2. 10. x = J or — }. 11. x = — J or 
-f. 12. x = 0or-4. 18. x=ior-f. 14. x= for -2. 18. x = a 

or - -. 16. X = ± - V^. 17. X = - or a. 18. x = a ± Vft. 19. x = a 

+ ft or 9lJlJ^, 20. X = 6 or -. 21. x = 6 or - a. 22. x = c or — . 

a -\- b b a 

28..x = 2a-6or -6. 24. x =- ^ ± J\/4a2 ^_ I2a + 1. 25. x = 2±v^. 

26. X = 1 ± a/^. 27. X = 2a or - 12. 28. x = § i W9-Sa^. 

29. X = 3 or J. 80. X = 5 or - ^%, 81. x = 36 or 12. 32. x = '^4 or 2. 

Page 256.-83. x = 6 or 21. 84. x = - V- ± |V^. 8 5. x = 4 

^^ 1 9ft ^ >. ^^ 7 •« ^ 22i:2V-3l9 ,a ^ 2 n ± V4 n^ - 3 
or — 1. 86. X = 4 or X. 87. x = — == • 88. x = = . 

^6 3 

89. X = a or — 1. 40. x = 13 a or — 6 o. 41. x = a or — 2 a. 

42. x = -± iVn2T4. 48. x = — ± — Vc^ ~4a6. 44. x=??^t^or 
2 2a 2a m — n 

2^=^. 46. x = w±n. 46. X = ^ ^ ^ » 47. x = 9ori. 48. x=lor 
n -f m 2c 

-L 49. x = 5i:2v^. 60. x=-^. 61. x = | or J. 62. x = 9 or 5. 
^ a-{-b 

68. x = 0. 64. x=#. 66. x = cor-. 66. x =--±- \/4o2 ^_ 9. 

^ c 2 2 

67. X = 10. 68. X = 4. 69. X = 6 or - V^. 60. x = 4 or 0. 61. x = 2 

or -A. 62. X = i ± tVv^3689. 63. x = ^5-±-^ or ^5-^. 64. x = 8 or 

-V- 66. x = ±V3. 66. x = Kl±V^)• 
Page 259. — 6. X = ± 3 or ± 1. 7. x = ± 5 or ± J\/3. 8. x_= 
±2>/2 or ±\/^. 9. x = 2 or \/^. 10. x = 2 or v^- 4. 
11. X = Vp ± Vg + p2. 12. a; = ± \/8 ; ± 2 or ± V^H". 18. x = i 1 or 
± \/f. 14.^ = 18 or 3. 16. x = 25 or 3. 16. x = ± y/2. 

17. X = ?^ly^. 18. x= if or 0. 19. x=4 or v^. 20. x = 4 or - 1. 
5 

21. x=3or v^^. 22. x=9or-2; i(7±vT73). 28. x=l±2Vl6orl. 



ANSWERS. 897 

84. x=:l(9±y/'^ ^) or j(3zfcV 7). 26. «= 13 or 78. 26. x=|f orO. 

27. x=V2a^±2aV€^-fb^+l>^. 28. x=±8or V(-Y)8. 29. a;=±l. 
80. a;=rt or J a. 

Page 260. — 81. a; =5 , 2, and 3. 32. x = 2, 2, and — 3. 88. x = - 1, 

I, 2, and — 2 ; or x = ± 1 and ±2, 34. x = 2, — 3, 1, and 4. 

Page 262.-6. 12 and 5. 7. 60 and 12. 8. 29 and 21. 9. 45 and 

62, or 52 and -45. 10. ^±iVm^-4:n] ^TiVm2-4n. 

2 ^ 

II. ^_|.iVw2n2_4w/t; -T— V^n^zriw^. 12. 9 or -11. 18. 18, 
2 2n 2 2n 

9, and 6. 

Page 263. — 14. Persons, 19; amount each paid, $10. 15. Breadth, 
34 rods ; length, 35 rods. 16. First square, 5625 men ; second square, 
7225 men. 17. 4 miles per hour. 18. 10 sheep. 19. Age, 36 yrs. 20. A, 
10 miles per hour ; B, 9 miles per hour. 21. 25 robes. 22. 118| sq. rds. 
23. 63 acres ; selling price, $ 50 per acre. 

Page 264. — 24. 15 pieces; cost of each, $84. 25. 12 persons. 
26. $80. 27. 12 horses. 28. One pipe, 15 hours; other pipe, 10 hours. 
15 miles per hour. 80. 600 lbs. 81. 84 sheep. 82. 20 cts. 



per dozen ; x ^ z:ac±V^l±^abc^ 

2a 

Page 266. — 2. x = 8 or — 5 ; y = 5 or — 8. 8. x = 5 or 12 J ; y = S 
or - llj. 4. X = 8 or J ; y = 3 or - ][. 5. x = 5 or 13f ; y = 4 or - 13J. 
6. X = 5 or — 4| J ; y = 4 or — 3J f . 7. x = 1 or J ; y = 2 or f . 8. x = 4 
or If? ; y = 5 or 8|^. 9. x = 2 or - i ; y = 4 or f . 10. x = 1 or 2^y ; 
y = 3 or 1^7-. 11. X = 3 or - I3S5 ; y = 2 or - 9^. 12. x = 5 or - 1^^ ; 
y = 6or-l|J. 

Page 267. —14. x=±3 or ±^V7 ; y=±2or ±\V1, 16. x=±l or 
ip V2 ; y = ± 2 or ± j V2. 16. x = ± 3 or T ? Vf ; y = ± 1 or ± f \/7. 
17. X = ± 4 or T f v/7 ; y = ± 2 or ± f \/7. 18. x = ± 6 or tW2j 
y = i 5 or ± Y>/^- 10- «= ±3 or =F |V-21; y = ±2 or ± ?V-21. 
20. x=±4 or ±2V3; y=±S or ± V3. 21. x=±3 or ±§v^; y=±2 
or ±\V2 . 22 . x=±3 or TJV^; y=zfc2 or ±jV2. 28. x=±8\/^ 
or db AV387I; y = ±V^- or ± iV387T. 

Page 269. — 27. x = 7 or — 5 ; y = 5 or — 7. 28. x = 5 or — 3 ; 
y = — 3 or 5. 29. x = 5 or — 6 ; y = — 6 or 5. 80. x = 3 or 2 ; y = 2 
or 3. 81. X = 4 or — 3 ; y = f or — 2. 82. x = 5 or — 2 ; y = 2 or —5. 
88. X = 4 or 3 ; y = 3 or 4. 84. x = 8 ; y = 5. 86. x = 5 or — 2 ; y = 2 
or —5. 86. X = 4 or —3 ; y = 3 or —4. 87. x = 6 or —5 ; y = — 5 or 6. 
88. X = 10 or — 5 ; y = 5 or —10. 

Page 270. — 4. x = 5 ; y = 3. 8. x = 9 ; y = 7. 6. x = 3 or — 2 ; 
y=2 or —3. 7. x=7 or 5; y=5 or 7. 8. x=±5; y=±3. 9. x=±6; 

y = ±2. 

Page 271. — 10. x = 5 or 3 ; y = 3 or 5. 11. x = 5 or - 2 ; y = - 2 
or 6. 12. i«J = 2 or 1 ; y = 1 or 2. X8. ap = 3 or 2 ; y = 2 or 3 ; x = 



i 



398 

4; y 

y = - 
aud 9. 

Pat 
13. L 

pipe, 
Pa 
P? 

S.x 

P 

3. r 

9:r 
9. 

I 
6. 



C-- 



4. 
6. 






ANSWER& 399 

Faiec 29a— 1. 1030. S. -336. S. 3G9. 4. 35. S. 237.5. 6. 5050. 

^ -^-^ 

Pa^^ 299. — 8. N'. 9. (in?; »; (m8)2n-l. 10.8=240. 11. d = -J. 
12. « = 143. 1S./=18.1. 14. a = 4-5. U. it=ll. iaa=-3;5 = 6. 
17. S444>. $48(j, $520, and $560. 18. 12 yards. 19. 140 days. 

90. 297110 yards. 

Page 30r-5. 128. a ^^I,^. 

Page 302.— 7. tsI^t- 8. a = 6. 9. a = 15. 10. r = 3. 11. r=f. 

12 2, J, 5, ?.S, If. 18. \ ^, ^, ^^ 1, V^, 3, 3v^, 9. 14. \ ?^, 
*"*'-' ^^ 9933 6 35 

28 8V7 m^ 32vj ^4^ ^ a ^z _ 

245' 245' 1715' 1715* 1715* ' x h •'' ^-^<^' 

Page 304.— 90. #=-29524. tl.s=\^4':l. 28.» = lf|i. 83. a =3. 
94. a = 1. 95. o = 8. 96. a = 979.2. 97. |.' 98. 2. 29. li. 80. 3i. 

81. 



n-1 

Page 305.-88. If. ^ 84.65;- 85. ^\\. 86. ilg^J. 87. 8|H. 
88. 54^;^. 

Page 306.-1- r =3 ; #=4372. 8. «=635 ; r=2. 8. 0=5^; » = 1365iJ. 
4. a = 2l; » = 40957J. 5. i = 1594323. 6. r = 2; n = 22. 7- « = 3. 

8. « =-T^. «- » = 221?. 10- 5. 1. I. y. V- 11. r = tV or 10%. 

12. 2, 10, 50. 18- - 3, - 18, - 108, - 648, or 21, - 12S, 77i, - 462f . 

14. $ 10,000. 15. 15,625,000,000 grains. 

Page 307. — 16- 891 1 feet ; 90 feet. 17. 3, 6, 12, 24, or - 24, - 12, 
-6, -3. 

Page 308.— 8. 3. 4. -^. 5. V and V- 6. J, A» A» 1. A. 

Page 316. — 5. 4^95424. 6. 3.92942. 7. 3.80821. 8. 4.50651. 
Page 317.— IL 3.65437. 12. 0.93725. 18. 1.89709. 14. 4.53866. 
Page 318.— 15. 2, 3, 4, 7, 9. 16. 30, 500, 6000, .008, .09. 17. 12, 

23, 45, 66, 84. 18. 136.249 ; 326.496 ; 597.493 ; 73.173 ; .93119. 
19. 4375.05 ; 59621.9 ; 673.22 ; 8.16056 ; .00053274. 

Page 323.-6. 182.97 + . 7. 4.42+. 8. 6.0115+. 9. 66.394. 

10. .00037658. 11. 6.0i:i8+. 12. .19114. 18. .000050962. 14. 19770. 

15. 1.7018. 16. .0036648. 17. 1.7489+. 18. 1.011. 

Page 324.— 19. n=8. 20. n=Q. 21. «=5. 22. n=5. 28. x=3. 

24. X = 3. 25. X = IJ, nearly. 26. x = 1.029. 27. x = 1.59+. 

28. j.^ ^ogc-^og<» 
log 6 

Page 325.— 2. 8%. 8. $450. 4. n= ^^^^ — 
^ /o V log(l+r) 

Page 328.-8. $8456.52. 4. $4856|. 5. $3322. 6. 23.98 yis. 
V. 23.44+ yrs. 8. $145985.20. 9. $1554.04. 10. $106.63. 

11. $125. 12. $27272/]-. 



400 ANSWERS. 

Page 332.-1. .J^ + -L^. 2. -§1 ??_. 3. -?-4-^i_ 

X — 6 ac — 8 X — 8 x — 7 x— 5 x—1 

4. ^—1-. 6. -^- + -^. 6. -L + ^-l. 7. 2 2 



x+5 x-2 3x+l 2x+l a+2- x+3 x a:-*-! 

+ ^_. 8.3 + -i_ + _!— ?. 9. -«- + -? !«-+ 2 



X — 1 X — 1 x+1 X X— 1 x + 1 2x — 1 2x -f- l' 

10.-^-+-?--?. ll.-i^-J_. 12.1+ 2 4 



x-3 x + 3 X x + 4 x + 2 5x~5 lOx+16' 

Page 334. —1. 1 - 6x+ 15x2 - 45x8 + 136x* . 2. l + 2x + 3x2 

+4x8+5x*+.... 3. l-fx+5x2+13x84-41x*4-121x5+.... 4. l+2x-fx2 
-x8-2x*-x6-f .... 6. 1 + ^x4- ]iV«^+ yVa^H ifi3«*H- •••• 6. 1 + 3x 

+ 7x2+ 17x8+41x*+99xs4- —. 7, 1 _ a; _ a;2+ 5x8- 7x*- x^H . 

8. a - (2 a - 6)x + (a - 2 6)x2 + (4 a + 6)x8 _ (H a - 4 6)x* + ..... 

Page338.-4. 1-1.^-+-^-^^ Ll^J* _^_ 5^^ 1 



4 16 4.8.162 4.8.12.168 3.9 



3 . 6 . 92 3 . 6 . 9 . 98 ' * 4.5 4 . 8 . 52 4 . 8 . 12 . 58 



8. 1+^- 



a a2 



2 2.4 



7. 2^(l+-i L:J— + ''^'^ V 

\ 3.2.4 3.6.22.42^3.6.9.28.48 ; 

I 3a8 f. ^ 2a 1-2 « 1.2.4 « ^^ ^-g 0^-41. 

2.4.6 3 3.6 3.6.9 

+6a-5&2_i0a-«&8+.... 11. a-|-f.?a"36+|^a"*&2+?.lll§o-V7^8+.... 



3 3.6 3.6.9 



12.1+1.1 LJ_+ 1-4.9 



5 5 5 . 10 . 52 5 . 10 . 15 . 58 

Page339. — 2. 3.036589+. 8. 2.024398+. 4. 2.0022248+. 

6. 3.0024648 + . 

Page 343. —2. x = 2; y = l. S. x = &; y = 2. 4. x = l;y=l. 

6. x = ^^ -^f ; y ^ a/ - cd ^ g 10-12=-2. 7. 76. 8. -16. 
ac — bd ac — hd 

Page 347.-9. x = 3; y = 2;z = l. 10. x = 6;y = 8;a = 10. 
11. x = 2; y = -2; z = 0. 12. x=l; y = 2; z = Z. 

Page 366.-4. 120. 6. 5040. 6. 325. 7. 120 (at round table 24). 
8. 6720. 9. 1.3699. 10. 34650. 

Page368. — 3. 126. 4. 84. 6. 16 (16 incl.-l). 6. 495. 7. 67. 
6. 1,392,300. 9. 36. 10. 63 in A ; 62 in B. 



■1 



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To avoid fine, this book should be returned on 
or before the date last stamped below 



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